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COMP 414/514: Optimization - Algorithms, Complexity and Approximations Lecture 1

COMP 414/514: Optimization - Algorithms, Complexity and

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Page 1: COMP 414/514: Optimization - Algorithms, Complexity and

COMP 414/514: Optimization - Algorithms, Complexity

and Approximations

Lecture 1

Page 2: COMP 414/514: Optimization - Algorithms, Complexity and

Overview

minx

f(x)

s.t. x 2 C<latexit sha1_base64="VS5+BwXu/958fDcsLNtKePYYGz8=">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</latexit><latexit sha1_base64="VS5+BwXu/958fDcsLNtKePYYGz8=">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</latexit><latexit sha1_base64="VS5+BwXu/958fDcsLNtKePYYGz8=">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</latexit><latexit sha1_base64="VS5+BwXu/958fDcsLNtKePYYGz8=">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</latexit>

Page 3: COMP 414/514: Optimization - Algorithms, Complexity and

Overview

minx

f(x)

s.t. x 2 C<latexit sha1_base64="VS5+BwXu/958fDcsLNtKePYYGz8=">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</latexit><latexit sha1_base64="VS5+BwXu/958fDcsLNtKePYYGz8=">AAACUnicbVJNaxsxEJWdNB9umjrpMRdRE5Nelt1QSI4mueSYQhwbLGO02llHRKtdpNlgI/Y3BkIu+SG59NBWaxvSxh0QPN57Mxo9FBdKWgzDl0ZzY/PD1vbObuvj3qf9z+2Dw1ubl0ZAX+QqN8OYW1BSQx8lKhgWBngWKxjE95e1PngAY2Wub3BewDjjUy1TKTh6atKWLIap1I4rOdWQVK0uZaVOfAegm1WOIczQZVJXtdSl6cnsG2Wsti0UG2CwVGaUSU1ZxvFOcOUuqxYDnbwNnrQ7YRAuiq6DaAU6ZFXXk/YTS3JRZqBRKG7tKAoLHDtuUAoFfnxpoeDink9h5KHmGdixW0RS0WPPJDTNjT8a6YL9u8PxzNp5FntnvbF9r9Xk/7RRien52EldlAhaLC9KS0Uxp3W+NJEGBKq5B1wY6Xel4o4bLtBnWocQvX/yOrg9DaIwiH587/QuVnHskCPylZyQiJyRHrki16RPBHkkr+QX+d14bvxs+l+ytDYbq54v5J9q7v0B3mOyfg==</latexit><latexit sha1_base64="VS5+BwXu/958fDcsLNtKePYYGz8=">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</latexit><latexit sha1_base64="VS5+BwXu/958fDcsLNtKePYYGz8=">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</latexit>

-Different objective classes -Different strategies within each problem -Different approaches based on computational capabilities —Different approaches based on constraints

Page 4: COMP 414/514: Optimization - Algorithms, Complexity and

Overview

minx

f(x)

s.t. x 2 C<latexit sha1_base64="VS5+BwXu/958fDcsLNtKePYYGz8=">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</latexit><latexit sha1_base64="VS5+BwXu/958fDcsLNtKePYYGz8=">AAACUnicbVJNaxsxEJWdNB9umjrpMRdRE5Nelt1QSI4mueSYQhwbLGO02llHRKtdpNlgI/Y3BkIu+SG59NBWaxvSxh0QPN57Mxo9FBdKWgzDl0ZzY/PD1vbObuvj3qf9z+2Dw1ubl0ZAX+QqN8OYW1BSQx8lKhgWBngWKxjE95e1PngAY2Wub3BewDjjUy1TKTh6atKWLIap1I4rOdWQVK0uZaVOfAegm1WOIczQZVJXtdSl6cnsG2Wsti0UG2CwVGaUSU1ZxvFOcOUuqxYDnbwNnrQ7YRAuiq6DaAU6ZFXXk/YTS3JRZqBRKG7tKAoLHDtuUAoFfnxpoeDink9h5KHmGdixW0RS0WPPJDTNjT8a6YL9u8PxzNp5FntnvbF9r9Xk/7RRien52EldlAhaLC9KS0Uxp3W+NJEGBKq5B1wY6Xel4o4bLtBnWocQvX/yOrg9DaIwiH587/QuVnHskCPylZyQiJyRHrki16RPBHkkr+QX+d14bvxs+l+ytDYbq54v5J9q7v0B3mOyfg==</latexit><latexit sha1_base64="VS5+BwXu/958fDcsLNtKePYYGz8=">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</latexit><latexit sha1_base64="VS5+BwXu/958fDcsLNtKePYYGz8=">AAACUnicbVJNaxsxEJWdNB9umjrpMRdRE5Nelt1QSI4mueSYQhwbLGO02llHRKtdpNlgI/Y3BkIu+SG59NBWaxvSxh0QPN57Mxo9FBdKWgzDl0ZzY/PD1vbObuvj3qf9z+2Dw1ubl0ZAX+QqN8OYW1BSQx8lKhgWBngWKxjE95e1PngAY2Wub3BewDjjUy1TKTh6atKWLIap1I4rOdWQVK0uZaVOfAegm1WOIczQZVJXtdSl6cnsG2Wsti0UG2CwVGaUSU1ZxvFOcOUuqxYDnbwNnrQ7YRAuiq6DaAU6ZFXXk/YTS3JRZqBRKG7tKAoLHDtuUAoFfnxpoeDink9h5KHmGdixW0RS0WPPJDTNjT8a6YL9u8PxzNp5FntnvbF9r9Xk/7RRien52EldlAhaLC9KS0Uxp3W+NJEGBKq5B1wY6Xel4o4bLtBnWocQvX/yOrg9DaIwiH587/QuVnHskCPylZyQiJyRHrki16RPBHkkr+QX+d14bvxs+l+ytDYbq54v5J9q7v0B3mOyfg==</latexit>

And, always having in mind applications in machine learning, AI and signal processing

-Different objective classes -Different strategies within each problem -Different approaches based on computational capabilities —Different approaches based on constraints

Page 5: COMP 414/514: Optimization - Algorithms, Complexity and

Motivation

Provable efficiency

(no fancy images included)

Page 6: COMP 414/514: Optimization - Algorithms, Complexity and

Motivation

Provable efficiency

Lots of data

(no fancy images included)

Page 7: COMP 414/514: Optimization - Algorithms, Complexity and

Motivation

Provable efficiency

Lots of data

Harder problems

(no fancy images included)

Page 8: COMP 414/514: Optimization - Algorithms, Complexity and

Motivation

Provable efficiency

Lots of data

Harder problems

More complicated models

(no fancy images included)

Page 9: COMP 414/514: Optimization - Algorithms, Complexity and

Motivation

Provable efficiency

Lots of data

Harder problems

More complicated models

Better results

(no fancy images included)

Page 10: COMP 414/514: Optimization - Algorithms, Complexity and

Motivation

Provable efficiency

Lots of data

Harder problems

More complicated models

Better results

(no fancy images included)

Page 11: COMP 414/514: Optimization - Algorithms, Complexity and

Motivation

Provable efficiency

Lots of data

Harder problems

More complicated models

Better results

(no fancy images included)

?

Page 12: COMP 414/514: Optimization - Algorithms, Complexity and

Motivation

Provable efficiency

(no fancy images included)

“What shall we do?”

Page 13: COMP 414/514: Optimization - Algorithms, Complexity and

Motivation

Provable efficiency

(no fancy images included)

Set up algo nicely Converge faster

Use prior knowledge Exploit resources

“What shall we do?”

Page 14: COMP 414/514: Optimization - Algorithms, Complexity and

Topics

- Continuous optimization (in general)

- Both theory and practice

- See syllabus

Page 15: COMP 414/514: Optimization - Algorithms, Complexity and

Topics

- Continuous optimization (in general)

- Both theory and practice

- Recent applications that drive research

- See syllabus

Page 16: COMP 414/514: Optimization - Algorithms, Complexity and

Topics

- Continuous optimization (in general)

- When no theory applies, some intuition

- Both theory and practice

- Recent applications that drive research

- See syllabus

Page 17: COMP 414/514: Optimization - Algorithms, Complexity and

Examples

- Least squares / linear regression(No, we will not re-define it)

minx

f(x)

s.t. x 2 C<latexit sha1_base64="VS5+BwXu/958fDcsLNtKePYYGz8=">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</latexit><latexit sha1_base64="VS5+BwXu/958fDcsLNtKePYYGz8=">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</latexit><latexit sha1_base64="VS5+BwXu/958fDcsLNtKePYYGz8=">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</latexit><latexit sha1_base64="VS5+BwXu/958fDcsLNtKePYYGz8=">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</latexit>

Page 18: COMP 414/514: Optimization - Algorithms, Complexity and

Examples

- Least squares / linear regression(No, we will not re-define it)

minx

f(x)

s.t. x 2 C<latexit sha1_base64="VS5+BwXu/958fDcsLNtKePYYGz8=">AAACUnicbVJNaxsxEJWdNB9umjrpMRdRE5Nelt1QSI4mueSYQhwbLGO02llHRKtdpNlgI/Y3BkIu+SG59NBWaxvSxh0QPN57Mxo9FBdKWgzDl0ZzY/PD1vbObuvj3qf9z+2Dw1ubl0ZAX+QqN8OYW1BSQx8lKhgWBngWKxjE95e1PngAY2Wub3BewDjjUy1TKTh6atKWLIap1I4rOdWQVK0uZaVOfAegm1WOIczQZVJXtdSl6cnsG2Wsti0UG2CwVGaUSU1ZxvFOcOUuqxYDnbwNnrQ7YRAuiq6DaAU6ZFXXk/YTS3JRZqBRKG7tKAoLHDtuUAoFfnxpoeDink9h5KHmGdixW0RS0WPPJDTNjT8a6YL9u8PxzNp5FntnvbF9r9Xk/7RRien52EldlAhaLC9KS0Uxp3W+NJEGBKq5B1wY6Xel4o4bLtBnWocQvX/yOrg9DaIwiH587/QuVnHskCPylZyQiJyRHrki16RPBHkkr+QX+d14bvxs+l+ytDYbq54v5J9q7v0B3mOyfg==</latexit><latexit sha1_base64="VS5+BwXu/958fDcsLNtKePYYGz8=">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</latexit><latexit sha1_base64="VS5+BwXu/958fDcsLNtKePYYGz8=">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</latexit><latexit sha1_base64="VS5+BwXu/958fDcsLNtKePYYGz8=">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</latexit>

)<latexit sha1_base64="uHo+69r8Q2XYAV1GqyfyNivJ6DE=">AAAB83icbVBNS8NAEJ34WetX1aOXxSJ4KkkV9Fj04rGK/YAmlM120y7d7IbdjVJC/4YXD4p49c9489+4aXPQ1gcDj/dmmJkXJpxp47rfzsrq2vrGZmmrvL2zu7dfOThsa5kqQltEcqm6IdaUM0FbhhlOu4miOA457YTjm9zvPFKlmRQPZpLQIMZDwSJGsLGS79+z4chgpeRTuV+pujV3BrRMvIJUoUCzX/nyB5KkMRWGcKx1z3MTE2RYGUY4nZb9VNMEkzEe0p6lAsdUB9ns5ik6tcoARVLZEgbN1N8TGY61nsSh7YyxGelFLxf/83qpia6CjIkkNVSQ+aIo5chIlAeABkxRYvjEEkwUs7ciMsIKE2NjykPwFl9eJu16zTuv1e8uqo3rIo4SHMMJnIEHl9CAW2hCCwgk8Ayv8Oakzovz7nzMW1ecYuYI/sD5/AHLDpGF</latexit>

minx

1n

nX

i=1

�yi � a>i x

�2

<latexit sha1_base64="XpXOqWlffPShbAZhVwS5E7lgniY=">AAACLXicbVDLSgMxFM3UV62vqks3wSLowjJTBd0IRV24VLBV6LRDJs20wSQzJHfEMswPufFXRHChiFt/w/SxUOuBCyfn3EvuPWEiuAHXfXMKM7Nz8wvFxdLS8srqWnl9o2niVFPWoLGI9W1IDBNcsQZwEOw20YzIULCb8O5s6N/cM214rK5hkLC2JD3FI04JWCkon/uSqyB7yLEPkSY08/JM2YdJZZBxfIK9vKOwL1gEu4OA431MAt7xIU7wg695rw97nVpQrrhVdwQ8TbwJqaAJLoPyi9+NaSqZAiqIMS3PTaCdEQ2cCpaX/NSwhNA70mMtSxWRzLSz0bU53rFKF0extqUAj9SfExmRxgxkaDslgb756w3F/7xWCtFxO+MqSYEpOv4oSgWGGA+jw12uGQUxsIRQze2umPaJDQ1swCUbgvf35GnSrFW9g2rt6rBSP53EUURbaBvtIg8doTq6QJeogSh6RM/oDb07T86r8+F8jlsLzmRmE/2C8/UN/F2n6g==</latexit>

Page 19: COMP 414/514: Optimization - Algorithms, Complexity and

Examples

- Quantum state tomography from limited samples

minX

f(X)

s.t. X 2 C<latexit sha1_base64="sGsk1R/o+2zG0t9KW1A5D6d1Ggg=">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</latexit>

Page 20: COMP 414/514: Optimization - Algorithms, Complexity and

Examples

- Quantum state tomography from limited samples

minX

f(X)

s.t. X 2 C<latexit sha1_base64="sGsk1R/o+2zG0t9KW1A5D6d1Ggg=">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</latexit>

)<latexit sha1_base64="uHo+69r8Q2XYAV1GqyfyNivJ6DE=">AAAB83icbVBNS8NAEJ34WetX1aOXxSJ4KkkV9Fj04rGK/YAmlM120y7d7IbdjVJC/4YXD4p49c9489+4aXPQ1gcDj/dmmJkXJpxp47rfzsrq2vrGZmmrvL2zu7dfOThsa5kqQltEcqm6IdaUM0FbhhlOu4miOA457YTjm9zvPFKlmRQPZpLQIMZDwSJGsLGS79+z4chgpeRTuV+pujV3BrRMvIJUoUCzX/nyB5KkMRWGcKx1z3MTE2RYGUY4nZb9VNMEkzEe0p6lAsdUB9ns5ik6tcoARVLZEgbN1N8TGY61nsSh7YyxGelFLxf/83qpia6CjIkkNVSQ+aIo5chIlAeABkxRYvjEEkwUs7ciMsIKE2NjykPwFl9eJu16zTuv1e8uqo3rIo4SHMMJnIEHl9CAW2hCCwgk8Ayv8Oakzovz7nzMW1ecYuYI/sD5/AHLDpGF</latexit>

minX

nX

i=1

�yi � Tr(A>

i X)�2

s.t. Tr(X) 1

X ⌫ 0<latexit sha1_base64="aGYUt/lWXQ/IAqu8kc3f+j+awrM=">AAACn3icbVHLjtMwFHXCayiPKbCEhaECdRZESUGCDdIwLEACoSJNZ4rqNnKcm9Qaxwn2DaKK8lt8CDv+BqeNeMxwLUtH55z78HVSKWkxDH96/qXLV65e27s+uHHz1u394Z27J7asjYCZKFVp5gm3oKSGGUpUMK8M8CJRcJqcven0069grCz1MW4qWBY81zKTgqOj4uF3lkAudcOVzDWk7eAJZbVOXQZgM28bhvANm0LqtpOcaOsibiR9RaN2pSlTkOF4E0v6lO6sx6Ydv47limFZ0fkBZUbmazxYTShjXfGtyQYY9PV+J3VeBV9otDN2Z961E8KR4YCBTv9MGQ9HYRBug14EUQ9GpI9pPPzB0lLUBWgUilu7iMIKlw03KIWCdsBqCxUXZzyHhYOaF2CXzXa/LX3smJRmpXFXI92yf2c0vLB2UyTOWXBc2/NaR/5PW9SYvVw2Ulc1gha7RlmtKJa0+yyaSgMC1cYBLox0s1Kx5oYLdB/ULSE6/+SL4GQSRM+Cyafno8Ojfh175D55RMYkIi/IIXlHpmRGhPfAO/Leex/8h/5b/6M/3Vl9r8+5R/4J//MvBZnHyw==</latexit>

Page 21: COMP 414/514: Optimization - Algorithms, Complexity and

Examples

- Fleet management

minX

f(X)

s.t. X 2 C<latexit sha1_base64="sGsk1R/o+2zG0t9KW1A5D6d1Ggg=">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</latexit>

Page 22: COMP 414/514: Optimization - Algorithms, Complexity and

Examples

- Fleet management

minX

f(X)

s.t. X 2 C<latexit sha1_base64="sGsk1R/o+2zG0t9KW1A5D6d1Ggg=">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</latexit>

)<latexit sha1_base64="uHo+69r8Q2XYAV1GqyfyNivJ6DE=">AAAB83icbVBNS8NAEJ34WetX1aOXxSJ4KkkV9Fj04rGK/YAmlM120y7d7IbdjVJC/4YXD4p49c9489+4aXPQ1gcDj/dmmJkXJpxp47rfzsrq2vrGZmmrvL2zu7dfOThsa5kqQltEcqm6IdaUM0FbhhlOu4miOA457YTjm9zvPFKlmRQPZpLQIMZDwSJGsLGS79+z4chgpeRTuV+pujV3BrRMvIJUoUCzX/nyB5KkMRWGcKx1z3MTE2RYGUY4nZb9VNMEkzEe0p6lAsdUB9ns5ik6tcoARVLZEgbN1N8TGY61nsSh7YyxGelFLxf/83qpia6CjIkkNVSQ+aIo5chIlAeABkxRYvjEEkwUs7ciMsIKE2NjykPwFl9eJu16zTuv1e8uqo3rIo4SHMMJnIEHl9CAW2hCCwgk8Ayv8Oakzovz7nzMW1ecYuYI/sD5/AHLDpGF</latexit>

minx2{0,1}m,y2{0,1}

f(y) =X

i2V

pX

k=1

di(1� q)qk�1yik

s.t.X

j2Wi

xj �pX

k=1

yik, i 2 V

X

j2Wxj = p

xj pj<latexit sha1_base64="jZsV8uBPKzNMytXFNyubTxVKwSI=">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</latexit>

Page 23: COMP 414/514: Optimization - Algorithms, Complexity and

Examples

- Neural networks

Input Layer ∈ ℝ²⁰ Hidden Layer ∈ ℝ¹² Hidden Layer ∈ ℝ¹⁰ Output Layer ∈ ℝ¹⁰

Page 24: COMP 414/514: Optimization - Algorithms, Complexity and

Examples

- Neural networks

)<latexit sha1_base64="uHo+69r8Q2XYAV1GqyfyNivJ6DE=">AAAB83icbVBNS8NAEJ34WetX1aOXxSJ4KkkV9Fj04rGK/YAmlM120y7d7IbdjVJC/4YXD4p49c9489+4aXPQ1gcDj/dmmJkXJpxp47rfzsrq2vrGZmmrvL2zu7dfOThsa5kqQltEcqm6IdaUM0FbhhlOu4miOA457YTjm9zvPFKlmRQPZpLQIMZDwSJGsLGS79+z4chgpeRTuV+pujV3BrRMvIJUoUCzX/nyB5KkMRWGcKx1z3MTE2RYGUY4nZb9VNMEkzEe0p6lAsdUB9ns5ik6tcoARVLZEgbN1N8TGY61nsSh7YyxGelFLxf/83qpia6CjIkkNVSQ+aIo5chIlAeABkxRYvjEEkwUs7ciMsIKE2NjykPwFl9eJu16zTuv1e8uqo3rIo4SHMMJnIEHl9CAW2hCCwgk8Ayv8Oakzovz7nzMW1ecYuYI/sD5/AHLDpGF</latexit>

minX

f(X)

s.t. X 2 C<latexit sha1_base64="sGsk1R/o+2zG0t9KW1A5D6d1Ggg=">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</latexit>

minWi

f(W1,W2) :=1n

nX

i=1

L (byi, yi) .<latexit sha1_base64="QNen4m0Bsq5tUS+0xdmpC7PONQY=">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</latexit>

where

byi = softmax (� (W2 · � (W1 · xi)))<latexit sha1_base64="ZcBWZy5z4ovkGqaxBKpDQY7s4ZA=">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</latexit>

Page 25: COMP 414/514: Optimization - Algorithms, Complexity and

Overview

minx

f(x)

s.t. x 2 C<latexit sha1_base64="VS5+BwXu/958fDcsLNtKePYYGz8=">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</latexit><latexit sha1_base64="VS5+BwXu/958fDcsLNtKePYYGz8=">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</latexit><latexit sha1_base64="VS5+BwXu/958fDcsLNtKePYYGz8=">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</latexit><latexit sha1_base64="VS5+BwXu/958fDcsLNtKePYYGz8=">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</latexit>

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Overview

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f(x)

s.t. x 2 C<latexit sha1_base64="VS5+BwXu/958fDcsLNtKePYYGz8=">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</latexit><latexit sha1_base64="VS5+BwXu/958fDcsLNtKePYYGz8=">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</latexit><latexit sha1_base64="VS5+BwXu/958fDcsLNtKePYYGz8=">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</latexit><latexit sha1_base64="VS5+BwXu/958fDcsLNtKePYYGz8=">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</latexit>

- The set of points that satisfy the constraintis called the feasible set

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Overview

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f(x)

s.t. x 2 C<latexit sha1_base64="VS5+BwXu/958fDcsLNtKePYYGz8=">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</latexit><latexit sha1_base64="VS5+BwXu/958fDcsLNtKePYYGz8=">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</latexit><latexit sha1_base64="VS5+BwXu/958fDcsLNtKePYYGz8=">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</latexit><latexit sha1_base64="VS5+BwXu/958fDcsLNtKePYYGz8=">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</latexit>

- The set of points that satisfy the constraintis called the feasible set

- Finding the point(s) that satisfies theconstraint and minimizes the objectiveis the task of optimization

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Overview

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f(x)

s.t. x 2 C<latexit sha1_base64="VS5+BwXu/958fDcsLNtKePYYGz8=">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</latexit><latexit sha1_base64="VS5+BwXu/958fDcsLNtKePYYGz8=">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</latexit><latexit sha1_base64="VS5+BwXu/958fDcsLNtKePYYGz8=">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</latexit><latexit sha1_base64="VS5+BwXu/958fDcsLNtKePYYGz8=">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</latexit>

- The set of points that satisfy the constraintis called the feasible set

- Finding the point(s) that satisfies theconstraint and minimizes the objectiveis the task of optimization

Unconstrained optimization

Page 29: COMP 414/514: Optimization - Algorithms, Complexity and

Disclaimer

Optimization is generally unsolvable..

(Closed form expressions vs. Iterative methods)

(Naive solvers that work well on specific cases)

Page 30: COMP 414/514: Optimization - Algorithms, Complexity and

This course focuses on iterative methods

- General procedure minx

f(x)

s.t. x 2 C<latexit sha1_base64="VS5+BwXu/958fDcsLNtKePYYGz8=">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</latexit><latexit sha1_base64="VS5+BwXu/958fDcsLNtKePYYGz8=">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</latexit><latexit sha1_base64="VS5+BwXu/958fDcsLNtKePYYGz8=">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</latexit><latexit sha1_base64="VS5+BwXu/958fDcsLNtKePYYGz8=">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</latexit>

Page 31: COMP 414/514: Optimization - Algorithms, Complexity and

This course focuses on iterative methods

- General procedure minx

f(x)

s.t. x 2 C<latexit sha1_base64="VS5+BwXu/958fDcsLNtKePYYGz8=">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</latexit><latexit sha1_base64="VS5+BwXu/958fDcsLNtKePYYGz8=">AAACUnicbVJNaxsxEJWdNB9umjrpMRdRE5Nelt1QSI4mueSYQhwbLGO02llHRKtdpNlgI/Y3BkIu+SG59NBWaxvSxh0QPN57Mxo9FBdKWgzDl0ZzY/PD1vbObuvj3qf9z+2Dw1ubl0ZAX+QqN8OYW1BSQx8lKhgWBngWKxjE95e1PngAY2Wub3BewDjjUy1TKTh6atKWLIap1I4rOdWQVK0uZaVOfAegm1WOIczQZVJXtdSl6cnsG2Wsti0UG2CwVGaUSU1ZxvFOcOUuqxYDnbwNnrQ7YRAuiq6DaAU6ZFXXk/YTS3JRZqBRKG7tKAoLHDtuUAoFfnxpoeDink9h5KHmGdixW0RS0WPPJDTNjT8a6YL9u8PxzNp5FntnvbF9r9Xk/7RRien52EldlAhaLC9KS0Uxp3W+NJEGBKq5B1wY6Xel4o4bLtBnWocQvX/yOrg9DaIwiH587/QuVnHskCPylZyQiJyRHrki16RPBHkkr+QX+d14bvxs+l+ytDYbq54v5J9q7v0B3mOyfg==</latexit><latexit sha1_base64="VS5+BwXu/958fDcsLNtKePYYGz8=">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</latexit><latexit sha1_base64="VS5+BwXu/958fDcsLNtKePYYGz8=">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</latexit>

- Key points that need to be addressed?

Page 32: COMP 414/514: Optimization - Algorithms, Complexity and

This course focuses on iterative methods

- General procedure minx

f(x)

s.t. x 2 C<latexit sha1_base64="VS5+BwXu/958fDcsLNtKePYYGz8=">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</latexit><latexit sha1_base64="VS5+BwXu/958fDcsLNtKePYYGz8=">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</latexit><latexit sha1_base64="VS5+BwXu/958fDcsLNtKePYYGz8=">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</latexit><latexit sha1_base64="VS5+BwXu/958fDcsLNtKePYYGz8=">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</latexit>

- Key points that need to be addressed?

- The notion of the Black-Box model

Page 33: COMP 414/514: Optimization - Algorithms, Complexity and

This course focuses on iterative methods

- General procedure minx

f(x)

s.t. x 2 C<latexit sha1_base64="VS5+BwXu/958fDcsLNtKePYYGz8=">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</latexit><latexit sha1_base64="VS5+BwXu/958fDcsLNtKePYYGz8=">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</latexit><latexit sha1_base64="VS5+BwXu/958fDcsLNtKePYYGz8=">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</latexit><latexit sha1_base64="VS5+BwXu/958fDcsLNtKePYYGz8=">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</latexit>

- Key points that need to be addressed?

- The notion of the Black-Box model

Page 34: COMP 414/514: Optimization - Algorithms, Complexity and

Topics NOT covered in this course

Page 35: COMP 414/514: Optimization - Algorithms, Complexity and

Topics NOT covered in this course

- (Mixed) integer programming

Page 36: COMP 414/514: Optimization - Algorithms, Complexity and

Topics NOT covered in this course

- (Mixed) integer programming

- Combinatorial optimization algorithms(E.g., Graph algorithms)

Page 37: COMP 414/514: Optimization - Algorithms, Complexity and

Topics NOT covered in this course

- (Mixed) integer programming

- Combinatorial optimization algorithms(E.g., Graph algorithms)

- Randomized algorithms (See Anshu’s course)

Page 38: COMP 414/514: Optimization - Algorithms, Complexity and

Topics NOT covered in this course

- (Mixed) integer programming

- Combinatorial optimization algorithms(E.g., Graph algorithms)

- Randomized algorithms (See Anshu’s course)

- Online algorithms, bandits

Page 39: COMP 414/514: Optimization - Algorithms, Complexity and

Topics NOT covered in this course

- (Mixed) integer programming

- Combinatorial optimization algorithms(E.g., Graph algorithms)

- Randomized algorithms (See Anshu’s course)

- Online algorithms, bandits

- Bayesian algorithms

Page 40: COMP 414/514: Optimization - Algorithms, Complexity and

Topics NOT covered in this course

- (Mixed) integer programming

- Combinatorial optimization algorithms(E.g., Graph algorithms)

- Randomized algorithms (See Anshu’s course)

- Online algorithms, bandits

- Bayesian algorithms

- Deep learning architectures (See Ankit’s course)

Page 41: COMP 414/514: Optimization - Algorithms, Complexity and

Any questions?

Page 42: COMP 414/514: Optimization - Algorithms, Complexity and

Who is this class for?

- Definitely, for PhD students

Page 43: COMP 414/514: Optimization - Algorithms, Complexity and

Who is this class for?

- Definitely, for PhD students

- Master or even undergraduates that want to start research(but get in touch with me soon to assess your background)

Page 44: COMP 414/514: Optimization - Algorithms, Complexity and

Who is this class for?

- Definitely, for PhD students

- Master or even undergraduates that want to start research(but get in touch with me soon to assess your background)

- Just auditing is fine by me

Page 45: COMP 414/514: Optimization - Algorithms, Complexity and

What is the vision for this course?

- For starters, this will always be an evolving course(Any feedback is more than welcome)

Page 46: COMP 414/514: Optimization - Algorithms, Complexity and

What is the vision for this course?

- For starters, this will always be an evolving course

- My purpose and vision is to introduce a series of optimization courses in the CS (and Duncan Hall’s in general) curriculum

(Any feedback is more than welcome)

Page 47: COMP 414/514: Optimization - Algorithms, Complexity and

What is the vision for this course?

- For starters, this will always be an evolving course

- My purpose and vision is to introduce a series of optimization courses in the CS (and Duncan Hall’s in general) curriculum

(Any feedback is more than welcome)

- The vision is for this course to be part of a sequence of courses that will focus on the theory+practice of methods

Page 48: COMP 414/514: Optimization - Algorithms, Complexity and

Course format

- Lectures (slides) + whiteboard + in-class code running(Some lectures have presentations, others will be handwritten)

Page 49: COMP 414/514: Optimization - Algorithms, Complexity and

Course format

- Lectures (slides) + whiteboard + in-class code running(Some lectures have presentations, others will be handwritten)

- Material broad enough to cover range of problems(But definitely will not cover each individual’s interest at 100%)

Page 50: COMP 414/514: Optimization - Algorithms, Complexity and

Course format

- Lectures (slides) + whiteboard + in-class code running

- Some material inspired from personal research

(Some lectures have presentations, others will be handwritten)

- Material broad enough to cover range of problems(But definitely will not cover each individual’s interest at 100%)

Page 51: COMP 414/514: Optimization - Algorithms, Complexity and

Course format

- Lectures (slides) + whiteboard + in-class code running

- Some material inspired from personal research

- “Interlude” lectures to provide some background (if needed)

(Some lectures have presentations, others will be handwritten)

- Material broad enough to cover range of problems(But definitely will not cover each individual’s interest at 100%)

Page 52: COMP 414/514: Optimization - Algorithms, Complexity and

Course format

- Lectures (slides) + whiteboard + in-class code running

- Some material inspired from personal research

- “Interlude” lectures to provide some background (if needed)

- Your workload: Graduate - HWs, final project Undergraduate - HWs, final exam

(Additional workload: scribing)

(Some lectures have presentations, others will be handwritten)

- Material broad enough to cover range of problems(But definitely will not cover each individual’s interest at 100%)

Page 53: COMP 414/514: Optimization - Algorithms, Complexity and

Regarding assignments

- Weakly assignments

Page 54: COMP 414/514: Optimization - Algorithms, Complexity and

Regarding assignments

- Weakly assignments

- Some questions are harder than others

Page 55: COMP 414/514: Optimization - Algorithms, Complexity and

Regarding assignments

- Weakly assignments

- Some questions are harder than others

- Some questions might not feel intuitive (“I’m a computer scientist! Why should I care about optimization?”)

Page 56: COMP 414/514: Optimization - Algorithms, Complexity and

Regarding assignments

- Weakly assignments

- Some questions are harder than others

- Some questions might not feel intuitive (“I’m a computer scientist! Why should I care about optimization?”)

- Try to do the best you can

(There will be a reweighing at the end of the course, only if necessary)

Page 57: COMP 414/514: Optimization - Algorithms, Complexity and

Goals + outcomes

- Learn about research in related fields

Page 58: COMP 414/514: Optimization - Algorithms, Complexity and

Goals + outcomes

- Learn about research in related fields

- Make connections between areas, understand how research advances in such areas..

Page 59: COMP 414/514: Optimization - Algorithms, Complexity and

Goals + outcomes

- Learn about research in related fields

- Make connections between areas, understand how research advances in such areas..

- Consider possible extensions of these works (project)

Page 60: COMP 414/514: Optimization - Algorithms, Complexity and

Goals + outcomes

- Learn about research in related fields

- Make connections between areas, understand how research advances in such areas..

- Consider possible extensions of these works (project)

- Comprehend how optimization is key in ML/AI/SP

Page 61: COMP 414/514: Optimization - Algorithms, Complexity and

Goals + outcomes

- Learn about research in related fields

- Make connections between areas, understand how research advances in such areas..

- Consider possible extensions of these works (project)

- Comprehend how optimization is key in ML/AI/SP

- Read and review recent papers

Page 62: COMP 414/514: Optimization - Algorithms, Complexity and

My goals

- Not to judge you on small details in HWs(But judge whether you have thought about solving the questions)

Page 63: COMP 414/514: Optimization - Algorithms, Complexity and

My goals

- Not to judge you on small details in HWs

- Spark your interest in research where math and practice are combined together

(But judge whether you have thought about solving the questions)

Page 64: COMP 414/514: Optimization - Algorithms, Complexity and

Prerequisites

- If you have taken any ML class, you are good to go

Page 65: COMP 414/514: Optimization - Algorithms, Complexity and

Prerequisites

- If you have taken any ML class, you are good to go

- Basics of calculus, linear algebra, basic knowledge of ML topics

Page 66: COMP 414/514: Optimization - Algorithms, Complexity and

Prerequisites

- If you have taken any ML class, you are good to go

- Basics of calculus, linear algebra, basic knowledge of ML topics

- Programming skills are not necessary(but might be required, depending on the project selected)

Page 67: COMP 414/514: Optimization - Algorithms, Complexity and

Prerequisites

- If you have taken any ML class, you are good to go

- Basics of calculus, linear algebra, basic knowledge of ML topics

- A quiz was usually provided for self-assessment, but I decided to make it an additional HW

- Programming skills are not necessary(but might be required, depending on the project selected)

Page 68: COMP 414/514: Optimization - Algorithms, Complexity and

Grading policy

- 45% HWs

- 50% project/final exam

- 5%: participation, attendance

- 5%: scribing notes (bonus)

Page 69: COMP 414/514: Optimization - Algorithms, Complexity and

Grading policy

- 45% HWs

- 50% project/final examUsually there is scaling in final grades. For me, a good grade is given based on the overall performance of the students: I value self-motivation, being proactive and enthusiasm.- 5%: participation, attendance

- 5%: scribing notes (bonus)

Page 70: COMP 414/514: Optimization - Algorithms, Complexity and

Participation + scribing

- “Παν μετρον αριστον” (Moderation is key)

Page 71: COMP 414/514: Optimization - Algorithms, Complexity and

Participation + scribing

- “Παν μετρον αριστον”

- Piazza might not be used - Questions over email is fine(I have set up an email for the course - see the syllabus)

(Moderation is key)

Page 72: COMP 414/514: Optimization - Algorithms, Complexity and

Participation + scribing

- “Παν μετρον αριστον”

- Piazza might not be used - Questions over email is fine(I have set up an email for the course - see the syllabus)

(Moderation is key)

- A slack channel will be set up

Page 73: COMP 414/514: Optimization - Algorithms, Complexity and

Participation + scribing

- “Παν μετρον αριστον”

- Piazza might not be used - Questions over email is fine(I have set up an email for the course - see the syllabus)

- Scribing is useful for you to understand better the material(or even get a better intuition than what was instructed)

(Moderation is key)

- A slack channel will be set up

Page 74: COMP 414/514: Optimization - Algorithms, Complexity and

Participation + scribing

- “Παν μετρον αριστον”

- Piazza might not be used - Questions over email is fine(I have set up an email for the course - see the syllabus)

- Scribing is useful for you to understand better the material(or even get a better intuition than what was instructed)

- Individuals or groups (2-3) of volunteers for each lecture(it will depend on the attendance)

(Moderation is key)

- A slack channel will be set up

Page 75: COMP 414/514: Optimization - Algorithms, Complexity and

Participation + scribing

- “Παν μετρον αριστον”

- Piazza might not be used - Questions over email is fine(I have set up an email for the course - see the syllabus)

- Scribing is useful for you to understand better the material(or even get a better intuition than what was instructed)

- Individuals or groups (2-3) of volunteers for each lecture(it will depend on the attendance)

- Deliverable in LaTEX(template available online)

(Moderation is key)

- A slack channel will be set up

Page 76: COMP 414/514: Optimization - Algorithms, Complexity and

HWs

- Deliverable in LaTEX

Page 77: COMP 414/514: Optimization - Algorithms, Complexity and

Reviews

- Select papers from a pile of .pdfs that will be provided(Reviews will be related to the topics currently taught)

(when applicable)

Page 78: COMP 414/514: Optimization - Algorithms, Complexity and

Reviews

- Single page reviews, similar to NIPS/ICML standards:

- Comment on novelty, clarity, importance

- Main comments + your overall score

(but not random as it usually is now)

- Select papers from a pile of .pdfs that will be provided(Reviews will be related to the topics currently taught)

(when applicable)

Page 79: COMP 414/514: Optimization - Algorithms, Complexity and

Presentations (for final projects)

- How does a project report look like?

Page 80: COMP 414/514: Optimization - Algorithms, Complexity and

Presentations (for final projects)

- How does a project report look like?

- Final exams: necessary for undergrads/optional for grads(?)

Page 81: COMP 414/514: Optimization - Algorithms, Complexity and

Presentations (for final projects)

- How does a project report look like?

- Final exams: necessary for undergrads/optional for grads(?)

- There might be some discussions during the lectures Take advantage by asking questions

(depending on the size of the class)

Page 82: COMP 414/514: Optimization - Algorithms, Complexity and

Presentations (for final projects)

- How does a project report look like?

- Final exams: necessary for undergrads/optional for grads(?)

- There might be some discussions during the lectures Take advantage by asking questions

(depending on the size of the class)

- Presentation should be at most XX minutes(tentative - depends on the class size)

Page 83: COMP 414/514: Optimization - Algorithms, Complexity and

Presentations (for final projects)

- How does a project report look like?

- Final exams: necessary for undergrads/optional for grads(?)

- There might be some discussions during the lectures Take advantage by asking questions

(depending on the size of the class)

- Grading: slides quality, clarity of main ideas

- Presentation should be at most XX minutes(tentative - depends on the class size)

Page 85: COMP 414/514: Optimization - Algorithms, Complexity and

Final Project

(Course website)

Page 86: COMP 414/514: Optimization - Algorithms, Complexity and

Final Project

(Course website)

Please come find me the earliest to discuss projects

Page 87: COMP 414/514: Optimization - Algorithms, Complexity and

Final Project

(Course website)

Please come find me the earliest to discuss projects

You should start reading papers soon, so that around mid-way you have a good project proposal

Page 88: COMP 414/514: Optimization - Algorithms, Complexity and

Communication

- I can handle emails in a very responsive way

Page 89: COMP 414/514: Optimization - Algorithms, Complexity and

Communication

- I can handle emails in a very responsive way

- Course email: [email protected] (please avoid sending emails to my personal account)

Page 90: COMP 414/514: Optimization - Algorithms, Complexity and

Communication

- I can handle emails in a very responsive way

- Course email: [email protected] (please avoid sending emails to my personal account)

- e-Mailing list: I will create one from registered students (So if you are not registered, you will not get updates)

Page 91: COMP 414/514: Optimization - Algorithms, Complexity and

Communication

- I can handle emails in a very responsive way

- Course email: [email protected] (please avoid sending emails to my personal account)

- e-Mailing list: I will create one from registered students (So if you are not registered, you will not get updates)

- HWs: will be sent to you via Canvas every week. (please do not distribute)

Page 92: COMP 414/514: Optimization - Algorithms, Complexity and

Notes

- I have started preparing notes for this course

Page 93: COMP 414/514: Optimization - Algorithms, Complexity and

Notes

- I have started preparing notes for this course

- (Bare with me - the current version was created while visiting coffee shops in Greece - so there will be typos, it definitely needs rephrasing at parts, references are missing, etc.)

Page 94: COMP 414/514: Optimization - Algorithms, Complexity and

Notes

- I have started preparing notes for this course

- (Bare with me - the current version was created while visiting coffee shops in Greece - so there will be typos, it definitely needs rephrasing at parts, references are missing, etc.)

- Every week I will try to upload an updated chapter; however I would appreciate any help with scribing throughout the semester

Page 95: COMP 414/514: Optimization - Algorithms, Complexity and

Length of each session

- There will be longer or shorter sessions

Page 96: COMP 414/514: Optimization - Algorithms, Complexity and

Length of each session

- There will be longer or shorter sessions

- Each week represents a Chapter

Page 97: COMP 414/514: Optimization - Algorithms, Complexity and

Length of each session

- There will be longer or shorter sessions

- Any feedback is more than welcome (e.g., too much material vs. too little material)

- Each week represents a Chapter

Page 98: COMP 414/514: Optimization - Algorithms, Complexity and

Length of each session

- There will be longer or shorter sessions

- Any feedback is more than welcome (e.g., too much material vs. too little material)

- In case I don’t have the time to cover fully a session, I will decide whether you will read it yourself, or I will teach it the next time.

- Each week represents a Chapter

Page 99: COMP 414/514: Optimization - Algorithms, Complexity and

Any questions?

Page 100: COMP 414/514: Optimization - Algorithms, Complexity and

Setting up the background

Page 101: COMP 414/514: Optimization - Algorithms, Complexity and

Vectors- Notation convention: vectors = lowercase, matrices = uppercase

Page 102: COMP 414/514: Optimization - Algorithms, Complexity and

Vectors- Notation convention: vectors = lowercase, matrices = uppercase

x =⇥x1, x2, . . . , xp

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Page 103: COMP 414/514: Optimization - Algorithms, Complexity and

Vectors- Notation convention: vectors = lowercase, matrices = uppercase

x =⇥x1, x2, . . . , xp

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- Vector in p-dimensions: x 2 Rp<latexit sha1_base64="pV6kYKYSFHH+vC+VFdq9vgk8zeE=">AAAB+3icbVDLSsNAFL2pr1pfsS7dDBbBVUlE0GXRjcsq9gFtLJPppB06mYSZibSE/IobF4q49Ufc+TdO2iy09cDA4Zx7uWeOH3OmtON8W6W19Y3NrfJ2ZWd3b//APqy2VZRIQlsk4pHs+lhRzgRtaaY57caS4tDntONPbnK/80SlYpF40LOYeiEeCRYwgrWRBnZ1ivpMoH6I9dj30/vsMR7YNafuzIFWiVuQGhRoDuyv/jAiSUiFJhwr1XOdWHsplpoRTrNKP1E0xmSCR7RnqMAhVV46z56hU6MMURBJ84RGc/X3RopDpWahbybzjGrZy8X/vF6igysvZSJONBVkcShIONIRyotAQyYp0XxmCCaSmayIjLHERJu6KqYEd/nLq6R9Xnedunt3UWtcF3WU4RhO4AxcuIQG3EITWkBgCs/wCm9WZr1Y79bHYrRkFTtH8AfW5w+izZQs</latexit><latexit sha1_base64="pV6kYKYSFHH+vC+VFdq9vgk8zeE=">AAAB+3icbVDLSsNAFL2pr1pfsS7dDBbBVUlE0GXRjcsq9gFtLJPppB06mYSZibSE/IobF4q49Ufc+TdO2iy09cDA4Zx7uWeOH3OmtON8W6W19Y3NrfJ2ZWd3b//APqy2VZRIQlsk4pHs+lhRzgRtaaY57caS4tDntONPbnK/80SlYpF40LOYeiEeCRYwgrWRBnZ1ivpMoH6I9dj30/vsMR7YNafuzIFWiVuQGhRoDuyv/jAiSUiFJhwr1XOdWHsplpoRTrNKP1E0xmSCR7RnqMAhVV46z56hU6MMURBJ84RGc/X3RopDpWahbybzjGrZy8X/vF6igysvZSJONBVkcShIONIRyotAQyYp0XxmCCaSmayIjLHERJu6KqYEd/nLq6R9Xnedunt3UWtcF3WU4RhO4AxcuIQG3EITWkBgCs/wCm9WZr1Y79bHYrRkFTtH8AfW5w+izZQs</latexit><latexit sha1_base64="pV6kYKYSFHH+vC+VFdq9vgk8zeE=">AAAB+3icbVDLSsNAFL2pr1pfsS7dDBbBVUlE0GXRjcsq9gFtLJPppB06mYSZibSE/IobF4q49Ufc+TdO2iy09cDA4Zx7uWeOH3OmtON8W6W19Y3NrfJ2ZWd3b//APqy2VZRIQlsk4pHs+lhRzgRtaaY57caS4tDntONPbnK/80SlYpF40LOYeiEeCRYwgrWRBnZ1ivpMoH6I9dj30/vsMR7YNafuzIFWiVuQGhRoDuyv/jAiSUiFJhwr1XOdWHsplpoRTrNKP1E0xmSCR7RnqMAhVV46z56hU6MMURBJ84RGc/X3RopDpWahbybzjGrZy8X/vF6igysvZSJONBVkcShIONIRyotAQyYp0XxmCCaSmayIjLHERJu6KqYEd/nLq6R9Xnedunt3UWtcF3WU4RhO4AxcuIQG3EITWkBgCs/wCm9WZr1Y79bHYrRkFTtH8AfW5w+izZQs</latexit><latexit sha1_base64="pV6kYKYSFHH+vC+VFdq9vgk8zeE=">AAAB+3icbVDLSsNAFL2pr1pfsS7dDBbBVUlE0GXRjcsq9gFtLJPppB06mYSZibSE/IobF4q49Ufc+TdO2iy09cDA4Zx7uWeOH3OmtON8W6W19Y3NrfJ2ZWd3b//APqy2VZRIQlsk4pHs+lhRzgRtaaY57caS4tDntONPbnK/80SlYpF40LOYeiEeCRYwgrWRBnZ1ivpMoH6I9dj30/vsMR7YNafuzIFWiVuQGhRoDuyv/jAiSUiFJhwr1XOdWHsplpoRTrNKP1E0xmSCR7RnqMAhVV46z56hU6MMURBJ84RGc/X3RopDpWahbybzjGrZy8X/vF6igysvZSJONBVkcShIONIRyotAQyYp0XxmCCaSmayIjLHERJu6KqYEd/nLq6R9Xnedunt3UWtcF3WU4RhO4AxcuIQG3EITWkBgCs/wCm9WZr1Y79bHYrRkFTtH8AfW5w+izZQs</latexit>

- Some properties of vectors:

x+ y = y + x, x, y 2 Rp<latexit sha1_base64="OFgRhEbB5Jrz4yAOoFxtg0Vizak=">AAACE3icbVDLSsNAFJ34rPUVdelmsAhipSQi6EYounFZxT6giWUymbZDJ5M4M5GG0H9w46+4caGIWzfu/BsnbRbaeuDC4Zx7ufceL2JUKsv6NubmFxaXlgsrxdW19Y1Nc2u7IcNYYFLHIQtFy0OSMMpJXVHFSCsSBAUeI01vcJn5zQciJA35rUoi4gaox2mXYqS01DEPh7AME3iuqwyHR9C5j5GfkQQ6lEMnQKrveenN6C7qmCWrYo0BZ4mdkxLIUeuYX44f4jggXGGGpGzbVqTcFAlFMSOjohNLEiE8QD3S1pSjgEg3Hf80gvta8WE3FLq4gmP190SKAimTwNOd2Y1y2svE/7x2rLpnbkp5FCvC8WRRN2ZQhTALCPpUEKxYognCgupbIe4jgbDSMRZ1CPb0y7OkcVyxrYp9fVKqXuRxFMAu2AMHwAanoAquQA3UAQaP4Bm8gjfjyXgx3o2PSeuckc/sgD8wPn8AmR+biQ==</latexit><latexit sha1_base64="OFgRhEbB5Jrz4yAOoFxtg0Vizak=">AAACE3icbVDLSsNAFJ34rPUVdelmsAhipSQi6EYounFZxT6giWUymbZDJ5M4M5GG0H9w46+4caGIWzfu/BsnbRbaeuDC4Zx7ufceL2JUKsv6NubmFxaXlgsrxdW19Y1Nc2u7IcNYYFLHIQtFy0OSMMpJXVHFSCsSBAUeI01vcJn5zQciJA35rUoi4gaox2mXYqS01DEPh7AME3iuqwyHR9C5j5GfkQQ6lEMnQKrveenN6C7qmCWrYo0BZ4mdkxLIUeuYX44f4jggXGGGpGzbVqTcFAlFMSOjohNLEiE8QD3S1pSjgEg3Hf80gvta8WE3FLq4gmP190SKAimTwNOd2Y1y2svE/7x2rLpnbkp5FCvC8WRRN2ZQhTALCPpUEKxYognCgupbIe4jgbDSMRZ1CPb0y7OkcVyxrYp9fVKqXuRxFMAu2AMHwAanoAquQA3UAQaP4Bm8gjfjyXgx3o2PSeuckc/sgD8wPn8AmR+biQ==</latexit><latexit sha1_base64="OFgRhEbB5Jrz4yAOoFxtg0Vizak=">AAACE3icbVDLSsNAFJ34rPUVdelmsAhipSQi6EYounFZxT6giWUymbZDJ5M4M5GG0H9w46+4caGIWzfu/BsnbRbaeuDC4Zx7ufceL2JUKsv6NubmFxaXlgsrxdW19Y1Nc2u7IcNYYFLHIQtFy0OSMMpJXVHFSCsSBAUeI01vcJn5zQciJA35rUoi4gaox2mXYqS01DEPh7AME3iuqwyHR9C5j5GfkQQ6lEMnQKrveenN6C7qmCWrYo0BZ4mdkxLIUeuYX44f4jggXGGGpGzbVqTcFAlFMSOjohNLEiE8QD3S1pSjgEg3Hf80gvta8WE3FLq4gmP190SKAimTwNOd2Y1y2svE/7x2rLpnbkp5FCvC8WRRN2ZQhTALCPpUEKxYognCgupbIe4jgbDSMRZ1CPb0y7OkcVyxrYp9fVKqXuRxFMAu2AMHwAanoAquQA3UAQaP4Bm8gjfjyXgx3o2PSeuckc/sgD8wPn8AmR+biQ==</latexit><latexit sha1_base64="OFgRhEbB5Jrz4yAOoFxtg0Vizak=">AAACE3icbVDLSsNAFJ34rPUVdelmsAhipSQi6EYounFZxT6giWUymbZDJ5M4M5GG0H9w46+4caGIWzfu/BsnbRbaeuDC4Zx7ufceL2JUKsv6NubmFxaXlgsrxdW19Y1Nc2u7IcNYYFLHIQtFy0OSMMpJXVHFSCsSBAUeI01vcJn5zQciJA35rUoi4gaox2mXYqS01DEPh7AME3iuqwyHR9C5j5GfkQQ6lEMnQKrveenN6C7qmCWrYo0BZ4mdkxLIUeuYX44f4jggXGGGpGzbVqTcFAlFMSOjohNLEiE8QD3S1pSjgEg3Hf80gvta8WE3FLq4gmP190SKAimTwNOd2Y1y2svE/7x2rLpnbkp5FCvC8WRRN2ZQhTALCPpUEKxYognCgupbIe4jgbDSMRZ1CPb0y7OkcVyxrYp9fVKqXuRxFMAu2AMHwAanoAquQA3UAQaP4Bm8gjfjyXgx3o2PSeuckc/sgD8wPn8AmR+biQ==</latexit>

(Commutative)

Page 104: COMP 414/514: Optimization - Algorithms, Complexity and

Vectors- Notation convention: vectors = lowercase, matrices = uppercase

x =⇥x1, x2, . . . , xp

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- Vector in p-dimensions: x 2 Rp<latexit sha1_base64="pV6kYKYSFHH+vC+VFdq9vgk8zeE=">AAAB+3icbVDLSsNAFL2pr1pfsS7dDBbBVUlE0GXRjcsq9gFtLJPppB06mYSZibSE/IobF4q49Ufc+TdO2iy09cDA4Zx7uWeOH3OmtON8W6W19Y3NrfJ2ZWd3b//APqy2VZRIQlsk4pHs+lhRzgRtaaY57caS4tDntONPbnK/80SlYpF40LOYeiEeCRYwgrWRBnZ1ivpMoH6I9dj30/vsMR7YNafuzIFWiVuQGhRoDuyv/jAiSUiFJhwr1XOdWHsplpoRTrNKP1E0xmSCR7RnqMAhVV46z56hU6MMURBJ84RGc/X3RopDpWahbybzjGrZy8X/vF6igysvZSJONBVkcShIONIRyotAQyYp0XxmCCaSmayIjLHERJu6KqYEd/nLq6R9Xnedunt3UWtcF3WU4RhO4AxcuIQG3EITWkBgCs/wCm9WZr1Y79bHYrRkFTtH8AfW5w+izZQs</latexit><latexit sha1_base64="pV6kYKYSFHH+vC+VFdq9vgk8zeE=">AAAB+3icbVDLSsNAFL2pr1pfsS7dDBbBVUlE0GXRjcsq9gFtLJPppB06mYSZibSE/IobF4q49Ufc+TdO2iy09cDA4Zx7uWeOH3OmtON8W6W19Y3NrfJ2ZWd3b//APqy2VZRIQlsk4pHs+lhRzgRtaaY57caS4tDntONPbnK/80SlYpF40LOYeiEeCRYwgrWRBnZ1ivpMoH6I9dj30/vsMR7YNafuzIFWiVuQGhRoDuyv/jAiSUiFJhwr1XOdWHsplpoRTrNKP1E0xmSCR7RnqMAhVV46z56hU6MMURBJ84RGc/X3RopDpWahbybzjGrZy8X/vF6igysvZSJONBVkcShIONIRyotAQyYp0XxmCCaSmayIjLHERJu6KqYEd/nLq6R9Xnedunt3UWtcF3WU4RhO4AxcuIQG3EITWkBgCs/wCm9WZr1Y79bHYrRkFTtH8AfW5w+izZQs</latexit><latexit sha1_base64="pV6kYKYSFHH+vC+VFdq9vgk8zeE=">AAAB+3icbVDLSsNAFL2pr1pfsS7dDBbBVUlE0GXRjcsq9gFtLJPppB06mYSZibSE/IobF4q49Ufc+TdO2iy09cDA4Zx7uWeOH3OmtON8W6W19Y3NrfJ2ZWd3b//APqy2VZRIQlsk4pHs+lhRzgRtaaY57caS4tDntONPbnK/80SlYpF40LOYeiEeCRYwgrWRBnZ1ivpMoH6I9dj30/vsMR7YNafuzIFWiVuQGhRoDuyv/jAiSUiFJhwr1XOdWHsplpoRTrNKP1E0xmSCR7RnqMAhVV46z56hU6MMURBJ84RGc/X3RopDpWahbybzjGrZy8X/vF6igysvZSJONBVkcShIONIRyotAQyYp0XxmCCaSmayIjLHERJu6KqYEd/nLq6R9Xnedunt3UWtcF3WU4RhO4AxcuIQG3EITWkBgCs/wCm9WZr1Y79bHYrRkFTtH8AfW5w+izZQs</latexit><latexit sha1_base64="pV6kYKYSFHH+vC+VFdq9vgk8zeE=">AAAB+3icbVDLSsNAFL2pr1pfsS7dDBbBVUlE0GXRjcsq9gFtLJPppB06mYSZibSE/IobF4q49Ufc+TdO2iy09cDA4Zx7uWeOH3OmtON8W6W19Y3NrfJ2ZWd3b//APqy2VZRIQlsk4pHs+lhRzgRtaaY57caS4tDntONPbnK/80SlYpF40LOYeiEeCRYwgrWRBnZ1ivpMoH6I9dj30/vsMR7YNafuzIFWiVuQGhRoDuyv/jAiSUiFJhwr1XOdWHsplpoRTrNKP1E0xmSCR7RnqMAhVV46z56hU6MMURBJ84RGc/X3RopDpWahbybzjGrZy8X/vF6igysvZSJONBVkcShIONIRyotAQyYp0XxmCCaSmayIjLHERJu6KqYEd/nLq6R9Xnedunt3UWtcF3WU4RhO4AxcuIQG3EITWkBgCs/wCm9WZr1Y79bHYrRkFTtH8AfW5w+izZQs</latexit>

- Some properties of vectors:

x+ y = y + x, x, y 2 Rp<latexit sha1_base64="OFgRhEbB5Jrz4yAOoFxtg0Vizak=">AAACE3icbVDLSsNAFJ34rPUVdelmsAhipSQi6EYounFZxT6giWUymbZDJ5M4M5GG0H9w46+4caGIWzfu/BsnbRbaeuDC4Zx7ufceL2JUKsv6NubmFxaXlgsrxdW19Y1Nc2u7IcNYYFLHIQtFy0OSMMpJXVHFSCsSBAUeI01vcJn5zQciJA35rUoi4gaox2mXYqS01DEPh7AME3iuqwyHR9C5j5GfkQQ6lEMnQKrveenN6C7qmCWrYo0BZ4mdkxLIUeuYX44f4jggXGGGpGzbVqTcFAlFMSOjohNLEiE8QD3S1pSjgEg3Hf80gvta8WE3FLq4gmP190SKAimTwNOd2Y1y2svE/7x2rLpnbkp5FCvC8WRRN2ZQhTALCPpUEKxYognCgupbIe4jgbDSMRZ1CPb0y7OkcVyxrYp9fVKqXuRxFMAu2AMHwAanoAquQA3UAQaP4Bm8gjfjyXgx3o2PSeuckc/sgD8wPn8AmR+biQ==</latexit><latexit sha1_base64="OFgRhEbB5Jrz4yAOoFxtg0Vizak=">AAACE3icbVDLSsNAFJ34rPUVdelmsAhipSQi6EYounFZxT6giWUymbZDJ5M4M5GG0H9w46+4caGIWzfu/BsnbRbaeuDC4Zx7ufceL2JUKsv6NubmFxaXlgsrxdW19Y1Nc2u7IcNYYFLHIQtFy0OSMMpJXVHFSCsSBAUeI01vcJn5zQciJA35rUoi4gaox2mXYqS01DEPh7AME3iuqwyHR9C5j5GfkQQ6lEMnQKrveenN6C7qmCWrYo0BZ4mdkxLIUeuYX44f4jggXGGGpGzbVqTcFAlFMSOjohNLEiE8QD3S1pSjgEg3Hf80gvta8WE3FLq4gmP190SKAimTwNOd2Y1y2svE/7x2rLpnbkp5FCvC8WRRN2ZQhTALCPpUEKxYognCgupbIe4jgbDSMRZ1CPb0y7OkcVyxrYp9fVKqXuRxFMAu2AMHwAanoAquQA3UAQaP4Bm8gjfjyXgx3o2PSeuckc/sgD8wPn8AmR+biQ==</latexit><latexit sha1_base64="OFgRhEbB5Jrz4yAOoFxtg0Vizak=">AAACE3icbVDLSsNAFJ34rPUVdelmsAhipSQi6EYounFZxT6giWUymbZDJ5M4M5GG0H9w46+4caGIWzfu/BsnbRbaeuDC4Zx7ufceL2JUKsv6NubmFxaXlgsrxdW19Y1Nc2u7IcNYYFLHIQtFy0OSMMpJXVHFSCsSBAUeI01vcJn5zQciJA35rUoi4gaox2mXYqS01DEPh7AME3iuqwyHR9C5j5GfkQQ6lEMnQKrveenN6C7qmCWrYo0BZ4mdkxLIUeuYX44f4jggXGGGpGzbVqTcFAlFMSOjohNLEiE8QD3S1pSjgEg3Hf80gvta8WE3FLq4gmP190SKAimTwNOd2Y1y2svE/7x2rLpnbkp5FCvC8WRRN2ZQhTALCPpUEKxYognCgupbIe4jgbDSMRZ1CPb0y7OkcVyxrYp9fVKqXuRxFMAu2AMHwAanoAquQA3UAQaP4Bm8gjfjyXgx3o2PSeuckc/sgD8wPn8AmR+biQ==</latexit><latexit sha1_base64="OFgRhEbB5Jrz4yAOoFxtg0Vizak=">AAACE3icbVDLSsNAFJ34rPUVdelmsAhipSQi6EYounFZxT6giWUymbZDJ5M4M5GG0H9w46+4caGIWzfu/BsnbRbaeuDC4Zx7ufceL2JUKsv6NubmFxaXlgsrxdW19Y1Nc2u7IcNYYFLHIQtFy0OSMMpJXVHFSCsSBAUeI01vcJn5zQciJA35rUoi4gaox2mXYqS01DEPh7AME3iuqwyHR9C5j5GfkQQ6lEMnQKrveenN6C7qmCWrYo0BZ4mdkxLIUeuYX44f4jggXGGGpGzbVqTcFAlFMSOjohNLEiE8QD3S1pSjgEg3Hf80gvta8WE3FLq4gmP190SKAimTwNOd2Y1y2svE/7x2rLpnbkp5FCvC8WRRN2ZQhTALCPpUEKxYognCgupbIe4jgbDSMRZ1CPb0y7OkcVyxrYp9fVKqXuRxFMAu2AMHwAanoAquQA3UAQaP4Bm8gjfjyXgx3o2PSeuckc/sgD8wPn8AmR+biQ==</latexit>

(Commutative)

(x+ y) + x = x+ (y + z), x, y, z 2 Rp<latexit sha1_base64="nwUlhc945BT21VgSiUDUslmkRMs=">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</latexit><latexit sha1_base64="nwUlhc945BT21VgSiUDUslmkRMs=">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</latexit><latexit sha1_base64="nwUlhc945BT21VgSiUDUslmkRMs=">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</latexit><latexit sha1_base64="nwUlhc945BT21VgSiUDUslmkRMs=">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</latexit>

(Associative)

Page 105: COMP 414/514: Optimization - Algorithms, Complexity and

Vectors- Notation convention: vectors = lowercase, matrices = uppercase

x =⇥x1, x2, . . . , xp

⇤><latexit sha1_base64="xUqRDgxO3ZplYL0R2xWoJkgk8h8=">AAACJHicbVDLSgMxFM34rPVVdekmWAQXpcyIoCBC0Y3LCrYKnbFk0tsazGSG5I60DP0YN/6KGxc+cOHGbzFTi2j1QMLhnHuTe0+YSGHQdd+dqemZ2bn5wkJxcWl5ZbW0tt40cao5NHgsY30ZMgNSKGigQAmXiQYWhRIuwpuT3L+4BW1ErM5xkEAQsZ4SXcEZWqldOuzTI+qH0BMqCyOGWvSHtN/2KvbarVC/E6PJeUJ9UJ3vkisf46RdKrtVdwT6l3hjUiZj1NulF/seTyNQyCUzpuW5CQYZ0yi4hGHRTw0kjN+wHrQsVSwCE2SjJYd02yod2o21PQrpSP3ZkbHImEEU2ko75LWZ9HLxP6+VYvcgyIRKUgTFvz7qppJiTPPEaEdo4CgHljCuhZ2V8mumGUeba9GG4E2u/Jc0d6ueW/XO9sq143EcBbJJtsgO8cg+qZFTUicNwskdeSBP5Nm5dx6dV+ftq3TKGfdskF9wPj4B3DmkXw==</latexit><latexit sha1_base64="xUqRDgxO3ZplYL0R2xWoJkgk8h8=">AAACJHicbVDLSgMxFM34rPVVdekmWAQXpcyIoCBC0Y3LCrYKnbFk0tsazGSG5I60DP0YN/6KGxc+cOHGbzFTi2j1QMLhnHuTe0+YSGHQdd+dqemZ2bn5wkJxcWl5ZbW0tt40cao5NHgsY30ZMgNSKGigQAmXiQYWhRIuwpuT3L+4BW1ErM5xkEAQsZ4SXcEZWqldOuzTI+qH0BMqCyOGWvSHtN/2KvbarVC/E6PJeUJ9UJ3vkisf46RdKrtVdwT6l3hjUiZj1NulF/seTyNQyCUzpuW5CQYZ0yi4hGHRTw0kjN+wHrQsVSwCE2SjJYd02yod2o21PQrpSP3ZkbHImEEU2ko75LWZ9HLxP6+VYvcgyIRKUgTFvz7qppJiTPPEaEdo4CgHljCuhZ2V8mumGUeba9GG4E2u/Jc0d6ueW/XO9sq143EcBbJJtsgO8cg+qZFTUicNwskdeSBP5Nm5dx6dV+ftq3TKGfdskF9wPj4B3DmkXw==</latexit><latexit sha1_base64="xUqRDgxO3ZplYL0R2xWoJkgk8h8=">AAACJHicbVDLSgMxFM34rPVVdekmWAQXpcyIoCBC0Y3LCrYKnbFk0tsazGSG5I60DP0YN/6KGxc+cOHGbzFTi2j1QMLhnHuTe0+YSGHQdd+dqemZ2bn5wkJxcWl5ZbW0tt40cao5NHgsY30ZMgNSKGigQAmXiQYWhRIuwpuT3L+4BW1ErM5xkEAQsZ4SXcEZWqldOuzTI+qH0BMqCyOGWvSHtN/2KvbarVC/E6PJeUJ9UJ3vkisf46RdKrtVdwT6l3hjUiZj1NulF/seTyNQyCUzpuW5CQYZ0yi4hGHRTw0kjN+wHrQsVSwCE2SjJYd02yod2o21PQrpSP3ZkbHImEEU2ko75LWZ9HLxP6+VYvcgyIRKUgTFvz7qppJiTPPEaEdo4CgHljCuhZ2V8mumGUeba9GG4E2u/Jc0d6ueW/XO9sq143EcBbJJtsgO8cg+qZFTUicNwskdeSBP5Nm5dx6dV+ftq3TKGfdskF9wPj4B3DmkXw==</latexit><latexit sha1_base64="xUqRDgxO3ZplYL0R2xWoJkgk8h8=">AAACJHicbVDLSgMxFM34rPVVdekmWAQXpcyIoCBC0Y3LCrYKnbFk0tsazGSG5I60DP0YN/6KGxc+cOHGbzFTi2j1QMLhnHuTe0+YSGHQdd+dqemZ2bn5wkJxcWl5ZbW0tt40cao5NHgsY30ZMgNSKGigQAmXiQYWhRIuwpuT3L+4BW1ErM5xkEAQsZ4SXcEZWqldOuzTI+qH0BMqCyOGWvSHtN/2KvbarVC/E6PJeUJ9UJ3vkisf46RdKrtVdwT6l3hjUiZj1NulF/seTyNQyCUzpuW5CQYZ0yi4hGHRTw0kjN+wHrQsVSwCE2SjJYd02yod2o21PQrpSP3ZkbHImEEU2ko75LWZ9HLxP6+VYvcgyIRKUgTFvz7qppJiTPPEaEdo4CgHljCuhZ2V8mumGUeba9GG4E2u/Jc0d6ueW/XO9sq143EcBbJJtsgO8cg+qZFTUicNwskdeSBP5Nm5dx6dV+ftq3TKGfdskF9wPj4B3DmkXw==</latexit>

- Vector in p-dimensions: x 2 Rp<latexit sha1_base64="pV6kYKYSFHH+vC+VFdq9vgk8zeE=">AAAB+3icbVDLSsNAFL2pr1pfsS7dDBbBVUlE0GXRjcsq9gFtLJPppB06mYSZibSE/IobF4q49Ufc+TdO2iy09cDA4Zx7uWeOH3OmtON8W6W19Y3NrfJ2ZWd3b//APqy2VZRIQlsk4pHs+lhRzgRtaaY57caS4tDntONPbnK/80SlYpF40LOYeiEeCRYwgrWRBnZ1ivpMoH6I9dj30/vsMR7YNafuzIFWiVuQGhRoDuyv/jAiSUiFJhwr1XOdWHsplpoRTrNKP1E0xmSCR7RnqMAhVV46z56hU6MMURBJ84RGc/X3RopDpWahbybzjGrZy8X/vF6igysvZSJONBVkcShIONIRyotAQyYp0XxmCCaSmayIjLHERJu6KqYEd/nLq6R9Xnedunt3UWtcF3WU4RhO4AxcuIQG3EITWkBgCs/wCm9WZr1Y79bHYrRkFTtH8AfW5w+izZQs</latexit><latexit sha1_base64="pV6kYKYSFHH+vC+VFdq9vgk8zeE=">AAAB+3icbVDLSsNAFL2pr1pfsS7dDBbBVUlE0GXRjcsq9gFtLJPppB06mYSZibSE/IobF4q49Ufc+TdO2iy09cDA4Zx7uWeOH3OmtON8W6W19Y3NrfJ2ZWd3b//APqy2VZRIQlsk4pHs+lhRzgRtaaY57caS4tDntONPbnK/80SlYpF40LOYeiEeCRYwgrWRBnZ1ivpMoH6I9dj30/vsMR7YNafuzIFWiVuQGhRoDuyv/jAiSUiFJhwr1XOdWHsplpoRTrNKP1E0xmSCR7RnqMAhVV46z56hU6MMURBJ84RGc/X3RopDpWahbybzjGrZy8X/vF6igysvZSJONBVkcShIONIRyotAQyYp0XxmCCaSmayIjLHERJu6KqYEd/nLq6R9Xnedunt3UWtcF3WU4RhO4AxcuIQG3EITWkBgCs/wCm9WZr1Y79bHYrRkFTtH8AfW5w+izZQs</latexit><latexit sha1_base64="pV6kYKYSFHH+vC+VFdq9vgk8zeE=">AAAB+3icbVDLSsNAFL2pr1pfsS7dDBbBVUlE0GXRjcsq9gFtLJPppB06mYSZibSE/IobF4q49Ufc+TdO2iy09cDA4Zx7uWeOH3OmtON8W6W19Y3NrfJ2ZWd3b//APqy2VZRIQlsk4pHs+lhRzgRtaaY57caS4tDntONPbnK/80SlYpF40LOYeiEeCRYwgrWRBnZ1ivpMoH6I9dj30/vsMR7YNafuzIFWiVuQGhRoDuyv/jAiSUiFJhwr1XOdWHsplpoRTrNKP1E0xmSCR7RnqMAhVV46z56hU6MMURBJ84RGc/X3RopDpWahbybzjGrZy8X/vF6igysvZSJONBVkcShIONIRyotAQyYp0XxmCCaSmayIjLHERJu6KqYEd/nLq6R9Xnedunt3UWtcF3WU4RhO4AxcuIQG3EITWkBgCs/wCm9WZr1Y79bHYrRkFTtH8AfW5w+izZQs</latexit><latexit sha1_base64="pV6kYKYSFHH+vC+VFdq9vgk8zeE=">AAAB+3icbVDLSsNAFL2pr1pfsS7dDBbBVUlE0GXRjcsq9gFtLJPppB06mYSZibSE/IobF4q49Ufc+TdO2iy09cDA4Zx7uWeOH3OmtON8W6W19Y3NrfJ2ZWd3b//APqy2VZRIQlsk4pHs+lhRzgRtaaY57caS4tDntONPbnK/80SlYpF40LOYeiEeCRYwgrWRBnZ1ivpMoH6I9dj30/vsMR7YNafuzIFWiVuQGhRoDuyv/jAiSUiFJhwr1XOdWHsplpoRTrNKP1E0xmSCR7RnqMAhVV46z56hU6MMURBJ84RGc/X3RopDpWahbybzjGrZy8X/vF6igysvZSJONBVkcShIONIRyotAQyYp0XxmCCaSmayIjLHERJu6KqYEd/nLq6R9Xnedunt3UWtcF3WU4RhO4AxcuIQG3EITWkBgCs/wCm9WZr1Y79bHYrRkFTtH8AfW5w+izZQs</latexit>

0 + x = x, x 2 Rp<latexit sha1_base64="k54SXwLVayyouAhdkPlS/CrKs1Q=">AAACDHicbVDLSgMxFL3js9ZX1aWbYBEEpcyIoBuh6MZlFfuATi2ZTKYNzWTGJCMtQz/Ajb/ixoUibv0Ad/6NmbYLbT0QODnnXu69x4s5U9q2v625+YXFpeXcSn51bX1js7C1XVNRIgmtkohHsuFhRTkTtKqZ5rQRS4pDj9O617vM/PoDlYpF4lYPYtoKcUewgBGsjdQuFG10iProHPWPkHufYN98XCaQG2Ld9bz0ZngXZ1UlewQ0S5wJKcIElXbhy/UjkoRUaMKxUk3HjnUrxVIzwukw7yaKxpj0cIc2DRU4pKqVjo4Zon2j+CiIpHlCo5H6uyPFoVKD0DOV2Y5q2svE/7xmooOzVspEnGgqyHhQkHCkI5Qlg3wmKdF8YAgmkpldEeliiYk2+eVNCM70ybOkdlxy7JJzfVIsX0ziyMEu7MEBOHAKZbiCClSBwCM8wyu8WU/Wi/VufYxL56xJzw78gfX5A1YfmVE=</latexit><latexit sha1_base64="k54SXwLVayyouAhdkPlS/CrKs1Q=">AAACDHicbVDLSgMxFL3js9ZX1aWbYBEEpcyIoBuh6MZlFfuATi2ZTKYNzWTGJCMtQz/Ajb/ixoUibv0Ad/6NmbYLbT0QODnnXu69x4s5U9q2v625+YXFpeXcSn51bX1js7C1XVNRIgmtkohHsuFhRTkTtKqZ5rQRS4pDj9O617vM/PoDlYpF4lYPYtoKcUewgBGsjdQuFG10iProHPWPkHufYN98XCaQG2Ld9bz0ZngXZ1UlewQ0S5wJKcIElXbhy/UjkoRUaMKxUk3HjnUrxVIzwukw7yaKxpj0cIc2DRU4pKqVjo4Zon2j+CiIpHlCo5H6uyPFoVKD0DOV2Y5q2svE/7xmooOzVspEnGgqyHhQkHCkI5Qlg3wmKdF8YAgmkpldEeliiYk2+eVNCM70ybOkdlxy7JJzfVIsX0ziyMEu7MEBOHAKZbiCClSBwCM8wyu8WU/Wi/VufYxL56xJzw78gfX5A1YfmVE=</latexit><latexit sha1_base64="k54SXwLVayyouAhdkPlS/CrKs1Q=">AAACDHicbVDLSgMxFL3js9ZX1aWbYBEEpcyIoBuh6MZlFfuATi2ZTKYNzWTGJCMtQz/Ajb/ixoUibv0Ad/6NmbYLbT0QODnnXu69x4s5U9q2v625+YXFpeXcSn51bX1js7C1XVNRIgmtkohHsuFhRTkTtKqZ5rQRS4pDj9O617vM/PoDlYpF4lYPYtoKcUewgBGsjdQuFG10iProHPWPkHufYN98XCaQG2Ld9bz0ZngXZ1UlewQ0S5wJKcIElXbhy/UjkoRUaMKxUk3HjnUrxVIzwukw7yaKxpj0cIc2DRU4pKqVjo4Zon2j+CiIpHlCo5H6uyPFoVKD0DOV2Y5q2svE/7xmooOzVspEnGgqyHhQkHCkI5Qlg3wmKdF8YAgmkpldEeliiYk2+eVNCM70ybOkdlxy7JJzfVIsX0ziyMEu7MEBOHAKZbiCClSBwCM8wyu8WU/Wi/VufYxL56xJzw78gfX5A1YfmVE=</latexit><latexit sha1_base64="k54SXwLVayyouAhdkPlS/CrKs1Q=">AAACDHicbVDLSgMxFL3js9ZX1aWbYBEEpcyIoBuh6MZlFfuATi2ZTKYNzWTGJCMtQz/Ajb/ixoUibv0Ad/6NmbYLbT0QODnnXu69x4s5U9q2v625+YXFpeXcSn51bX1js7C1XVNRIgmtkohHsuFhRTkTtKqZ5rQRS4pDj9O617vM/PoDlYpF4lYPYtoKcUewgBGsjdQuFG10iProHPWPkHufYN98XCaQG2Ld9bz0ZngXZ1UlewQ0S5wJKcIElXbhy/UjkoRUaMKxUk3HjnUrxVIzwukw7yaKxpj0cIc2DRU4pKqVjo4Zon2j+CiIpHlCo5H6uyPFoVKD0DOV2Y5q2svE/7xmooOzVspEnGgqyHhQkHCkI5Qlg3wmKdF8YAgmkpldEeliiYk2+eVNCM70ybOkdlxy7JJzfVIsX0ziyMEu7MEBOHAKZbiCClSBwCM8wyu8WU/Wi/VufYxL56xJzw78gfX5A1YfmVE=</latexit>

- Some properties of vectors:

x+ y = y + x, x, y 2 Rp<latexit sha1_base64="OFgRhEbB5Jrz4yAOoFxtg0Vizak=">AAACE3icbVDLSsNAFJ34rPUVdelmsAhipSQi6EYounFZxT6giWUymbZDJ5M4M5GG0H9w46+4caGIWzfu/BsnbRbaeuDC4Zx7ufceL2JUKsv6NubmFxaXlgsrxdW19Y1Nc2u7IcNYYFLHIQtFy0OSMMpJXVHFSCsSBAUeI01vcJn5zQciJA35rUoi4gaox2mXYqS01DEPh7AME3iuqwyHR9C5j5GfkQQ6lEMnQKrveenN6C7qmCWrYo0BZ4mdkxLIUeuYX44f4jggXGGGpGzbVqTcFAlFMSOjohNLEiE8QD3S1pSjgEg3Hf80gvta8WE3FLq4gmP190SKAimTwNOd2Y1y2svE/7x2rLpnbkp5FCvC8WRRN2ZQhTALCPpUEKxYognCgupbIe4jgbDSMRZ1CPb0y7OkcVyxrYp9fVKqXuRxFMAu2AMHwAanoAquQA3UAQaP4Bm8gjfjyXgx3o2PSeuckc/sgD8wPn8AmR+biQ==</latexit><latexit sha1_base64="OFgRhEbB5Jrz4yAOoFxtg0Vizak=">AAACE3icbVDLSsNAFJ34rPUVdelmsAhipSQi6EYounFZxT6giWUymbZDJ5M4M5GG0H9w46+4caGIWzfu/BsnbRbaeuDC4Zx7ufceL2JUKsv6NubmFxaXlgsrxdW19Y1Nc2u7IcNYYFLHIQtFy0OSMMpJXVHFSCsSBAUeI01vcJn5zQciJA35rUoi4gaox2mXYqS01DEPh7AME3iuqwyHR9C5j5GfkQQ6lEMnQKrveenN6C7qmCWrYo0BZ4mdkxLIUeuYX44f4jggXGGGpGzbVqTcFAlFMSOjohNLEiE8QD3S1pSjgEg3Hf80gvta8WE3FLq4gmP190SKAimTwNOd2Y1y2svE/7x2rLpnbkp5FCvC8WRRN2ZQhTALCPpUEKxYognCgupbIe4jgbDSMRZ1CPb0y7OkcVyxrYp9fVKqXuRxFMAu2AMHwAanoAquQA3UAQaP4Bm8gjfjyXgx3o2PSeuckc/sgD8wPn8AmR+biQ==</latexit><latexit sha1_base64="OFgRhEbB5Jrz4yAOoFxtg0Vizak=">AAACE3icbVDLSsNAFJ34rPUVdelmsAhipSQi6EYounFZxT6giWUymbZDJ5M4M5GG0H9w46+4caGIWzfu/BsnbRbaeuDC4Zx7ufceL2JUKsv6NubmFxaXlgsrxdW19Y1Nc2u7IcNYYFLHIQtFy0OSMMpJXVHFSCsSBAUeI01vcJn5zQciJA35rUoi4gaox2mXYqS01DEPh7AME3iuqwyHR9C5j5GfkQQ6lEMnQKrveenN6C7qmCWrYo0BZ4mdkxLIUeuYX44f4jggXGGGpGzbVqTcFAlFMSOjohNLEiE8QD3S1pSjgEg3Hf80gvta8WE3FLq4gmP190SKAimTwNOd2Y1y2svE/7x2rLpnbkp5FCvC8WRRN2ZQhTALCPpUEKxYognCgupbIe4jgbDSMRZ1CPb0y7OkcVyxrYp9fVKqXuRxFMAu2AMHwAanoAquQA3UAQaP4Bm8gjfjyXgx3o2PSeuckc/sgD8wPn8AmR+biQ==</latexit><latexit sha1_base64="OFgRhEbB5Jrz4yAOoFxtg0Vizak=">AAACE3icbVDLSsNAFJ34rPUVdelmsAhipSQi6EYounFZxT6giWUymbZDJ5M4M5GG0H9w46+4caGIWzfu/BsnbRbaeuDC4Zx7ufceL2JUKsv6NubmFxaXlgsrxdW19Y1Nc2u7IcNYYFLHIQtFy0OSMMpJXVHFSCsSBAUeI01vcJn5zQciJA35rUoi4gaox2mXYqS01DEPh7AME3iuqwyHR9C5j5GfkQQ6lEMnQKrveenN6C7qmCWrYo0BZ4mdkxLIUeuYX44f4jggXGGGpGzbVqTcFAlFMSOjohNLEiE8QD3S1pSjgEg3Hf80gvta8WE3FLq4gmP190SKAimTwNOd2Y1y2svE/7x2rLpnbkp5FCvC8WRRN2ZQhTALCPpUEKxYognCgupbIe4jgbDSMRZ1CPb0y7OkcVyxrYp9fVKqXuRxFMAu2AMHwAanoAquQA3UAQaP4Bm8gjfjyXgx3o2PSeuckc/sgD8wPn8AmR+biQ==</latexit>

(Commutative)

(x+ y) + x = x+ (y + z), x, y, z 2 Rp<latexit sha1_base64="nwUlhc945BT21VgSiUDUslmkRMs=">AAACInicbVBLSwMxEM76tr5WPXoZLELFUnZFUA9C0YvHKvYB3Vqyaaqh2eyaZKXr0t/ixb/ixYOingR/jNm2B18fZPjyzQwz8/kRZ0o7zoc1MTk1PTM7N59bWFxaXrFX12oqjCWhVRLyUDZ8rChnglY105w2Iklx4HNa93snWb5+S6ViobjQSURbAb4SrMsI1kZq24eFPuxAsm1CH44g+xQSE+62i+DdxLgD/SIkRbgDjwnwAqyvfT89H1xGbTvvlJwh4C9xxySPxqi07TevE5I4oEITjpVquk6kWymWmhFOBzkvVjTCpIevaNNQgQOqWunwxAFsGaUD3VCaJzQM1e8dKQ6USgLfVGY7qt+5TPwv14x196CVMhHFmgoyGtSNOegQMr+gwyQlmieGYCKZ2RXINZaYaONqzpjg/j75L6ntllyn5J7t5cvHYzvm0AbaRAXkon1URqeogqqIoHv0iJ7Ri/VgPVmv1vuodMIa96yjH7A+vwA+y59P</latexit><latexit sha1_base64="nwUlhc945BT21VgSiUDUslmkRMs=">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</latexit><latexit sha1_base64="nwUlhc945BT21VgSiUDUslmkRMs=">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</latexit><latexit sha1_base64="nwUlhc945BT21VgSiUDUslmkRMs=">AAACInicbVBLSwMxEM76tr5WPXoZLELFUnZFUA9C0YvHKvYB3Vqyaaqh2eyaZKXr0t/ixb/ixYOingR/jNm2B18fZPjyzQwz8/kRZ0o7zoc1MTk1PTM7N59bWFxaXrFX12oqjCWhVRLyUDZ8rChnglY105w2Iklx4HNa93snWb5+S6ViobjQSURbAb4SrMsI1kZq24eFPuxAsm1CH44g+xQSE+62i+DdxLgD/SIkRbgDjwnwAqyvfT89H1xGbTvvlJwh4C9xxySPxqi07TevE5I4oEITjpVquk6kWymWmhFOBzkvVjTCpIevaNNQgQOqWunwxAFsGaUD3VCaJzQM1e8dKQ6USgLfVGY7qt+5TPwv14x196CVMhHFmgoyGtSNOegQMr+gwyQlmieGYCKZ2RXINZaYaONqzpjg/j75L6ntllyn5J7t5cvHYzvm0AbaRAXkon1URqeogqqIoHv0iJ7Ri/VgPVmv1vuodMIa96yjH7A+vwA+y59P</latexit>

(Associative)

Page 106: COMP 414/514: Optimization - Algorithms, Complexity and

Vectors- Notation convention: vectors = lowercase, matrices = uppercase

x =⇥x1, x2, . . . , xp

⇤><latexit sha1_base64="xUqRDgxO3ZplYL0R2xWoJkgk8h8=">AAACJHicbVDLSgMxFM34rPVVdekmWAQXpcyIoCBC0Y3LCrYKnbFk0tsazGSG5I60DP0YN/6KGxc+cOHGbzFTi2j1QMLhnHuTe0+YSGHQdd+dqemZ2bn5wkJxcWl5ZbW0tt40cao5NHgsY30ZMgNSKGigQAmXiQYWhRIuwpuT3L+4BW1ErM5xkEAQsZ4SXcEZWqldOuzTI+qH0BMqCyOGWvSHtN/2KvbarVC/E6PJeUJ9UJ3vkisf46RdKrtVdwT6l3hjUiZj1NulF/seTyNQyCUzpuW5CQYZ0yi4hGHRTw0kjN+wHrQsVSwCE2SjJYd02yod2o21PQrpSP3ZkbHImEEU2ko75LWZ9HLxP6+VYvcgyIRKUgTFvz7qppJiTPPEaEdo4CgHljCuhZ2V8mumGUeba9GG4E2u/Jc0d6ueW/XO9sq143EcBbJJtsgO8cg+qZFTUicNwskdeSBP5Nm5dx6dV+ftq3TKGfdskF9wPj4B3DmkXw==</latexit><latexit sha1_base64="xUqRDgxO3ZplYL0R2xWoJkgk8h8=">AAACJHicbVDLSgMxFM34rPVVdekmWAQXpcyIoCBC0Y3LCrYKnbFk0tsazGSG5I60DP0YN/6KGxc+cOHGbzFTi2j1QMLhnHuTe0+YSGHQdd+dqemZ2bn5wkJxcWl5ZbW0tt40cao5NHgsY30ZMgNSKGigQAmXiQYWhRIuwpuT3L+4BW1ErM5xkEAQsZ4SXcEZWqldOuzTI+qH0BMqCyOGWvSHtN/2KvbarVC/E6PJeUJ9UJ3vkisf46RdKrtVdwT6l3hjUiZj1NulF/seTyNQyCUzpuW5CQYZ0yi4hGHRTw0kjN+wHrQsVSwCE2SjJYd02yod2o21PQrpSP3ZkbHImEEU2ko75LWZ9HLxP6+VYvcgyIRKUgTFvz7qppJiTPPEaEdo4CgHljCuhZ2V8mumGUeba9GG4E2u/Jc0d6ueW/XO9sq143EcBbJJtsgO8cg+qZFTUicNwskdeSBP5Nm5dx6dV+ftq3TKGfdskF9wPj4B3DmkXw==</latexit><latexit sha1_base64="xUqRDgxO3ZplYL0R2xWoJkgk8h8=">AAACJHicbVDLSgMxFM34rPVVdekmWAQXpcyIoCBC0Y3LCrYKnbFk0tsazGSG5I60DP0YN/6KGxc+cOHGbzFTi2j1QMLhnHuTe0+YSGHQdd+dqemZ2bn5wkJxcWl5ZbW0tt40cao5NHgsY30ZMgNSKGigQAmXiQYWhRIuwpuT3L+4BW1ErM5xkEAQsZ4SXcEZWqldOuzTI+qH0BMqCyOGWvSHtN/2KvbarVC/E6PJeUJ9UJ3vkisf46RdKrtVdwT6l3hjUiZj1NulF/seTyNQyCUzpuW5CQYZ0yi4hGHRTw0kjN+wHrQsVSwCE2SjJYd02yod2o21PQrpSP3ZkbHImEEU2ko75LWZ9HLxP6+VYvcgyIRKUgTFvz7qppJiTPPEaEdo4CgHljCuhZ2V8mumGUeba9GG4E2u/Jc0d6ueW/XO9sq143EcBbJJtsgO8cg+qZFTUicNwskdeSBP5Nm5dx6dV+ftq3TKGfdskF9wPj4B3DmkXw==</latexit><latexit sha1_base64="xUqRDgxO3ZplYL0R2xWoJkgk8h8=">AAACJHicbVDLSgMxFM34rPVVdekmWAQXpcyIoCBC0Y3LCrYKnbFk0tsazGSG5I60DP0YN/6KGxc+cOHGbzFTi2j1QMLhnHuTe0+YSGHQdd+dqemZ2bn5wkJxcWl5ZbW0tt40cao5NHgsY30ZMgNSKGigQAmXiQYWhRIuwpuT3L+4BW1ErM5xkEAQsZ4SXcEZWqldOuzTI+qH0BMqCyOGWvSHtN/2KvbarVC/E6PJeUJ9UJ3vkisf46RdKrtVdwT6l3hjUiZj1NulF/seTyNQyCUzpuW5CQYZ0yi4hGHRTw0kjN+wHrQsVSwCE2SjJYd02yod2o21PQrpSP3ZkbHImEEU2ko75LWZ9HLxP6+VYvcgyIRKUgTFvz7qppJiTPPEaEdo4CgHljCuhZ2V8mumGUeba9GG4E2u/Jc0d6ueW/XO9sq143EcBbJJtsgO8cg+qZFTUicNwskdeSBP5Nm5dx6dV+ftq3TKGfdskF9wPj4B3DmkXw==</latexit>

- Vector in p-dimensions: x 2 Rp<latexit sha1_base64="pV6kYKYSFHH+vC+VFdq9vgk8zeE=">AAAB+3icbVDLSsNAFL2pr1pfsS7dDBbBVUlE0GXRjcsq9gFtLJPppB06mYSZibSE/IobF4q49Ufc+TdO2iy09cDA4Zx7uWeOH3OmtON8W6W19Y3NrfJ2ZWd3b//APqy2VZRIQlsk4pHs+lhRzgRtaaY57caS4tDntONPbnK/80SlYpF40LOYeiEeCRYwgrWRBnZ1ivpMoH6I9dj30/vsMR7YNafuzIFWiVuQGhRoDuyv/jAiSUiFJhwr1XOdWHsplpoRTrNKP1E0xmSCR7RnqMAhVV46z56hU6MMURBJ84RGc/X3RopDpWahbybzjGrZy8X/vF6igysvZSJONBVkcShIONIRyotAQyYp0XxmCCaSmayIjLHERJu6KqYEd/nLq6R9Xnedunt3UWtcF3WU4RhO4AxcuIQG3EITWkBgCs/wCm9WZr1Y79bHYrRkFTtH8AfW5w+izZQs</latexit><latexit sha1_base64="pV6kYKYSFHH+vC+VFdq9vgk8zeE=">AAAB+3icbVDLSsNAFL2pr1pfsS7dDBbBVUlE0GXRjcsq9gFtLJPppB06mYSZibSE/IobF4q49Ufc+TdO2iy09cDA4Zx7uWeOH3OmtON8W6W19Y3NrfJ2ZWd3b//APqy2VZRIQlsk4pHs+lhRzgRtaaY57caS4tDntONPbnK/80SlYpF40LOYeiEeCRYwgrWRBnZ1ivpMoH6I9dj30/vsMR7YNafuzIFWiVuQGhRoDuyv/jAiSUiFJhwr1XOdWHsplpoRTrNKP1E0xmSCR7RnqMAhVV46z56hU6MMURBJ84RGc/X3RopDpWahbybzjGrZy8X/vF6igysvZSJONBVkcShIONIRyotAQyYp0XxmCCaSmayIjLHERJu6KqYEd/nLq6R9Xnedunt3UWtcF3WU4RhO4AxcuIQG3EITWkBgCs/wCm9WZr1Y79bHYrRkFTtH8AfW5w+izZQs</latexit><latexit sha1_base64="pV6kYKYSFHH+vC+VFdq9vgk8zeE=">AAAB+3icbVDLSsNAFL2pr1pfsS7dDBbBVUlE0GXRjcsq9gFtLJPppB06mYSZibSE/IobF4q49Ufc+TdO2iy09cDA4Zx7uWeOH3OmtON8W6W19Y3NrfJ2ZWd3b//APqy2VZRIQlsk4pHs+lhRzgRtaaY57caS4tDntONPbnK/80SlYpF40LOYeiEeCRYwgrWRBnZ1ivpMoH6I9dj30/vsMR7YNafuzIFWiVuQGhRoDuyv/jAiSUiFJhwr1XOdWHsplpoRTrNKP1E0xmSCR7RnqMAhVV46z56hU6MMURBJ84RGc/X3RopDpWahbybzjGrZy8X/vF6igysvZSJONBVkcShIONIRyotAQyYp0XxmCCaSmayIjLHERJu6KqYEd/nLq6R9Xnedunt3UWtcF3WU4RhO4AxcuIQG3EITWkBgCs/wCm9WZr1Y79bHYrRkFTtH8AfW5w+izZQs</latexit><latexit sha1_base64="pV6kYKYSFHH+vC+VFdq9vgk8zeE=">AAAB+3icbVDLSsNAFL2pr1pfsS7dDBbBVUlE0GXRjcsq9gFtLJPppB06mYSZibSE/IobF4q49Ufc+TdO2iy09cDA4Zx7uWeOH3OmtON8W6W19Y3NrfJ2ZWd3b//APqy2VZRIQlsk4pHs+lhRzgRtaaY57caS4tDntONPbnK/80SlYpF40LOYeiEeCRYwgrWRBnZ1ivpMoH6I9dj30/vsMR7YNafuzIFWiVuQGhRoDuyv/jAiSUiFJhwr1XOdWHsplpoRTrNKP1E0xmSCR7RnqMAhVV46z56hU6MMURBJ84RGc/X3RopDpWahbybzjGrZy8X/vF6igysvZSJONBVkcShIONIRyotAQyYp0XxmCCaSmayIjLHERJu6KqYEd/nLq6R9Xnedunt3UWtcF3WU4RhO4AxcuIQG3EITWkBgCs/wCm9WZr1Y79bHYrRkFTtH8AfW5w+izZQs</latexit>

0 + x = x, x 2 Rp<latexit sha1_base64="k54SXwLVayyouAhdkPlS/CrKs1Q=">AAACDHicbVDLSgMxFL3js9ZX1aWbYBEEpcyIoBuh6MZlFfuATi2ZTKYNzWTGJCMtQz/Ajb/ixoUibv0Ad/6NmbYLbT0QODnnXu69x4s5U9q2v625+YXFpeXcSn51bX1js7C1XVNRIgmtkohHsuFhRTkTtKqZ5rQRS4pDj9O617vM/PoDlYpF4lYPYtoKcUewgBGsjdQuFG10iProHPWPkHufYN98XCaQG2Ld9bz0ZngXZ1UlewQ0S5wJKcIElXbhy/UjkoRUaMKxUk3HjnUrxVIzwukw7yaKxpj0cIc2DRU4pKqVjo4Zon2j+CiIpHlCo5H6uyPFoVKD0DOV2Y5q2svE/7xmooOzVspEnGgqyHhQkHCkI5Qlg3wmKdF8YAgmkpldEeliiYk2+eVNCM70ybOkdlxy7JJzfVIsX0ziyMEu7MEBOHAKZbiCClSBwCM8wyu8WU/Wi/VufYxL56xJzw78gfX5A1YfmVE=</latexit><latexit sha1_base64="k54SXwLVayyouAhdkPlS/CrKs1Q=">AAACDHicbVDLSgMxFL3js9ZX1aWbYBEEpcyIoBuh6MZlFfuATi2ZTKYNzWTGJCMtQz/Ajb/ixoUibv0Ad/6NmbYLbT0QODnnXu69x4s5U9q2v625+YXFpeXcSn51bX1js7C1XVNRIgmtkohHsuFhRTkTtKqZ5rQRS4pDj9O617vM/PoDlYpF4lYPYtoKcUewgBGsjdQuFG10iProHPWPkHufYN98XCaQG2Ld9bz0ZngXZ1UlewQ0S5wJKcIElXbhy/UjkoRUaMKxUk3HjnUrxVIzwukw7yaKxpj0cIc2DRU4pKqVjo4Zon2j+CiIpHlCo5H6uyPFoVKD0DOV2Y5q2svE/7xmooOzVspEnGgqyHhQkHCkI5Qlg3wmKdF8YAgmkpldEeliiYk2+eVNCM70ybOkdlxy7JJzfVIsX0ziyMEu7MEBOHAKZbiCClSBwCM8wyu8WU/Wi/VufYxL56xJzw78gfX5A1YfmVE=</latexit><latexit sha1_base64="k54SXwLVayyouAhdkPlS/CrKs1Q=">AAACDHicbVDLSgMxFL3js9ZX1aWbYBEEpcyIoBuh6MZlFfuATi2ZTKYNzWTGJCMtQz/Ajb/ixoUibv0Ad/6NmbYLbT0QODnnXu69x4s5U9q2v625+YXFpeXcSn51bX1js7C1XVNRIgmtkohHsuFhRTkTtKqZ5rQRS4pDj9O617vM/PoDlYpF4lYPYtoKcUewgBGsjdQuFG10iProHPWPkHufYN98XCaQG2Ld9bz0ZngXZ1UlewQ0S5wJKcIElXbhy/UjkoRUaMKxUk3HjnUrxVIzwukw7yaKxpj0cIc2DRU4pKqVjo4Zon2j+CiIpHlCo5H6uyPFoVKD0DOV2Y5q2svE/7xmooOzVspEnGgqyHhQkHCkI5Qlg3wmKdF8YAgmkpldEeliiYk2+eVNCM70ybOkdlxy7JJzfVIsX0ziyMEu7MEBOHAKZbiCClSBwCM8wyu8WU/Wi/VufYxL56xJzw78gfX5A1YfmVE=</latexit><latexit sha1_base64="k54SXwLVayyouAhdkPlS/CrKs1Q=">AAACDHicbVDLSgMxFL3js9ZX1aWbYBEEpcyIoBuh6MZlFfuATi2ZTKYNzWTGJCMtQz/Ajb/ixoUibv0Ad/6NmbYLbT0QODnnXu69x4s5U9q2v625+YXFpeXcSn51bX1js7C1XVNRIgmtkohHsuFhRTkTtKqZ5rQRS4pDj9O617vM/PoDlYpF4lYPYtoKcUewgBGsjdQuFG10iProHPWPkHufYN98XCaQG2Ld9bz0ZngXZ1UlewQ0S5wJKcIElXbhy/UjkoRUaMKxUk3HjnUrxVIzwukw7yaKxpj0cIc2DRU4pKqVjo4Zon2j+CiIpHlCo5H6uyPFoVKD0DOV2Y5q2svE/7xmooOzVspEnGgqyHhQkHCkI5Qlg3wmKdF8YAgmkpldEeliiYk2+eVNCM70ybOkdlxy7JJzfVIsX0ziyMEu7MEBOHAKZbiCClSBwCM8wyu8WU/Wi/VufYxL56xJzw78gfX5A1YfmVE=</latexit>

- Some properties of vectors:

x+ y = y + x, x, y 2 Rp<latexit sha1_base64="OFgRhEbB5Jrz4yAOoFxtg0Vizak=">AAACE3icbVDLSsNAFJ34rPUVdelmsAhipSQi6EYounFZxT6giWUymbZDJ5M4M5GG0H9w46+4caGIWzfu/BsnbRbaeuDC4Zx7ufceL2JUKsv6NubmFxaXlgsrxdW19Y1Nc2u7IcNYYFLHIQtFy0OSMMpJXVHFSCsSBAUeI01vcJn5zQciJA35rUoi4gaox2mXYqS01DEPh7AME3iuqwyHR9C5j5GfkQQ6lEMnQKrveenN6C7qmCWrYo0BZ4mdkxLIUeuYX44f4jggXGGGpGzbVqTcFAlFMSOjohNLEiE8QD3S1pSjgEg3Hf80gvta8WE3FLq4gmP190SKAimTwNOd2Y1y2svE/7x2rLpnbkp5FCvC8WRRN2ZQhTALCPpUEKxYognCgupbIe4jgbDSMRZ1CPb0y7OkcVyxrYp9fVKqXuRxFMAu2AMHwAanoAquQA3UAQaP4Bm8gjfjyXgx3o2PSeuckc/sgD8wPn8AmR+biQ==</latexit><latexit sha1_base64="OFgRhEbB5Jrz4yAOoFxtg0Vizak=">AAACE3icbVDLSsNAFJ34rPUVdelmsAhipSQi6EYounFZxT6giWUymbZDJ5M4M5GG0H9w46+4caGIWzfu/BsnbRbaeuDC4Zx7ufceL2JUKsv6NubmFxaXlgsrxdW19Y1Nc2u7IcNYYFLHIQtFy0OSMMpJXVHFSCsSBAUeI01vcJn5zQciJA35rUoi4gaox2mXYqS01DEPh7AME3iuqwyHR9C5j5GfkQQ6lEMnQKrveenN6C7qmCWrYo0BZ4mdkxLIUeuYX44f4jggXGGGpGzbVqTcFAlFMSOjohNLEiE8QD3S1pSjgEg3Hf80gvta8WE3FLq4gmP190SKAimTwNOd2Y1y2svE/7x2rLpnbkp5FCvC8WRRN2ZQhTALCPpUEKxYognCgupbIe4jgbDSMRZ1CPb0y7OkcVyxrYp9fVKqXuRxFMAu2AMHwAanoAquQA3UAQaP4Bm8gjfjyXgx3o2PSeuckc/sgD8wPn8AmR+biQ==</latexit><latexit sha1_base64="OFgRhEbB5Jrz4yAOoFxtg0Vizak=">AAACE3icbVDLSsNAFJ34rPUVdelmsAhipSQi6EYounFZxT6giWUymbZDJ5M4M5GG0H9w46+4caGIWzfu/BsnbRbaeuDC4Zx7ufceL2JUKsv6NubmFxaXlgsrxdW19Y1Nc2u7IcNYYFLHIQtFy0OSMMpJXVHFSCsSBAUeI01vcJn5zQciJA35rUoi4gaox2mXYqS01DEPh7AME3iuqwyHR9C5j5GfkQQ6lEMnQKrveenN6C7qmCWrYo0BZ4mdkxLIUeuYX44f4jggXGGGpGzbVqTcFAlFMSOjohNLEiE8QD3S1pSjgEg3Hf80gvta8WE3FLq4gmP190SKAimTwNOd2Y1y2svE/7x2rLpnbkp5FCvC8WRRN2ZQhTALCPpUEKxYognCgupbIe4jgbDSMRZ1CPb0y7OkcVyxrYp9fVKqXuRxFMAu2AMHwAanoAquQA3UAQaP4Bm8gjfjyXgx3o2PSeuckc/sgD8wPn8AmR+biQ==</latexit><latexit sha1_base64="OFgRhEbB5Jrz4yAOoFxtg0Vizak=">AAACE3icbVDLSsNAFJ34rPUVdelmsAhipSQi6EYounFZxT6giWUymbZDJ5M4M5GG0H9w46+4caGIWzfu/BsnbRbaeuDC4Zx7ufceL2JUKsv6NubmFxaXlgsrxdW19Y1Nc2u7IcNYYFLHIQtFy0OSMMpJXVHFSCsSBAUeI01vcJn5zQciJA35rUoi4gaox2mXYqS01DEPh7AME3iuqwyHR9C5j5GfkQQ6lEMnQKrveenN6C7qmCWrYo0BZ4mdkxLIUeuYX44f4jggXGGGpGzbVqTcFAlFMSOjohNLEiE8QD3S1pSjgEg3Hf80gvta8WE3FLq4gmP190SKAimTwNOd2Y1y2svE/7x2rLpnbkp5FCvC8WRRN2ZQhTALCPpUEKxYognCgupbIe4jgbDSMRZ1CPb0y7OkcVyxrYp9fVKqXuRxFMAu2AMHwAanoAquQA3UAQaP4Bm8gjfjyXgx3o2PSeuckc/sgD8wPn8AmR+biQ==</latexit>

(Commutative)

(x+ y) + x = x+ (y + z), x, y, z 2 Rp<latexit sha1_base64="nwUlhc945BT21VgSiUDUslmkRMs=">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</latexit><latexit sha1_base64="nwUlhc945BT21VgSiUDUslmkRMs=">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</latexit><latexit sha1_base64="nwUlhc945BT21VgSiUDUslmkRMs=">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</latexit><latexit sha1_base64="nwUlhc945BT21VgSiUDUslmkRMs=">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</latexit>

(Associative)

↵(x+ y) = ↵x+ ↵y, x, y 2 Rp<latexit sha1_base64="big65NoSvqjuoWkF8ZK4nskFE2s=">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</latexit><latexit sha1_base64="big65NoSvqjuoWkF8ZK4nskFE2s=">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</latexit><latexit sha1_base64="big65NoSvqjuoWkF8ZK4nskFE2s=">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</latexit><latexit sha1_base64="big65NoSvqjuoWkF8ZK4nskFE2s=">AAACK3icbVDLSsNAFJ34rPVVdenmYhEUpSQi6EaQunGpYlVoarmZTu3QySTOTMQQ+j9u/BUXuvCBW//DSc3C14GBM+fcy733BLHg2rjuqzMyOjY+MVmaKk/PzM7NVxYWz3SUKMoaNBKRughQM8ElaxhuBLuIFcMwEOw86B/k/vkNU5pH8tSkMWuFeCV5l1M0VmpX6j6KuIewBrewAek67EGh5P+CppvgXyfYgdtNSMHnEvwQTS8IspPBZdyuVN2aOwT8JV5BqqTAUbvy6HcimoRMGipQ66bnxqaVoTKcCjYo+4lmMdI+XrGmpRJDplvZ8NYBrFqlA91I2ScNDNXvHRmGWqdhYCvzHfVvLxf/85qJ6e62Mi7jxDBJvwZ1EwEmgjw46HDFqBGpJUgVt7sC7aFCamy8ZRuC9/vkv+Rsq+a5Ne94u7pfL+IokWWyQtaIR3bIPjkkR6RBKLkjD+SZvDj3zpPz5rx/lY44Rc8S+QHn4xPHHqRw</latexit>

(Distributive)

Page 107: COMP 414/514: Optimization - Algorithms, Complexity and

Vectors- Span of a set of vectors:

span {x1, x2, . . . , xk} = {↵1x1 + ↵2x2 + · · ·+ ↵kxk | ↵i 2 R, i = [1, k]}<latexit sha1_base64="jYzScHcJllG/zV8+LZ5t1Rg0umU=">AAAChHicbVHfb9MwEHYCjFF+FXjk5USFQKKq4sK0vWya4IXHgeg2qY4ix3VaK44T2RfUKsv+Ef4r3vhvcLpMwMZJlr777r67811aaeUwin4F4Z2793bu7z4YPHz0+MnT4bPnp66srZAzUerSnqfcSa2MnKFCLc8rK3mRanmW5p+6+Nl3aZ0qzTfcVDIu+NKoTAmOnkqGPxjKNTau4qZlWmbImnVCx7BOpmNgixJdh3Nm1XKFrIVD6LMY19WKJ9RHKbyD3p12ws7tlH/ovKsBlxeX14QCpgywguMqTZuv7RiULz33jfMYrpslw1E0ibYGtwHtwYj0dpIMf/q+oi6kQaG5c3MaVRg33KISWrYDVjtZcZHzpZx7aHghXdxsl9jCa88sICutfwZhy/6taHjh3KZIfWY3trsZ68j/xeY1Zgdxo0xVozTiqlFWa8ASuovAQlkpUG884MIqPyuIFbdcoL/bwC+B3vzybXA6ndBoQr98GB1/7NexS16SV+QtoWSfHJPP5ITMiAiC4E0QBTTcCcfh+3DvKjUMes0L8o+FR78BezS/Lw==</latexit><latexit sha1_base64="jYzScHcJllG/zV8+LZ5t1Rg0umU=">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</latexit><latexit sha1_base64="jYzScHcJllG/zV8+LZ5t1Rg0umU=">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</latexit><latexit sha1_base64="jYzScHcJllG/zV8+LZ5t1Rg0umU=">AAAChHicbVHfb9MwEHYCjFF+FXjk5USFQKKq4sK0vWya4IXHgeg2qY4ix3VaK44T2RfUKsv+Ef4r3vhvcLpMwMZJlr777r67811aaeUwin4F4Z2793bu7z4YPHz0+MnT4bPnp66srZAzUerSnqfcSa2MnKFCLc8rK3mRanmW5p+6+Nl3aZ0qzTfcVDIu+NKoTAmOnkqGPxjKNTau4qZlWmbImnVCx7BOpmNgixJdh3Nm1XKFrIVD6LMY19WKJ9RHKbyD3p12ws7tlH/ovKsBlxeX14QCpgywguMqTZuv7RiULz33jfMYrpslw1E0ibYGtwHtwYj0dpIMf/q+oi6kQaG5c3MaVRg33KISWrYDVjtZcZHzpZx7aHghXdxsl9jCa88sICutfwZhy/6taHjh3KZIfWY3trsZ68j/xeY1Zgdxo0xVozTiqlFWa8ASuovAQlkpUG884MIqPyuIFbdcoL/bwC+B3vzybXA6ndBoQr98GB1/7NexS16SV+QtoWSfHJPP5ITMiAiC4E0QBTTcCcfh+3DvKjUMes0L8o+FR78BezS/Lw==</latexit>

Page 108: COMP 414/514: Optimization - Algorithms, Complexity and

Vectors- Span of a set of vectors:

span {x1, x2, . . . , xk} = {↵1x1 + ↵2x2 + · · ·+ ↵kxk | ↵i 2 R, i = [1, k]}<latexit sha1_base64="jYzScHcJllG/zV8+LZ5t1Rg0umU=">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</latexit><latexit sha1_base64="jYzScHcJllG/zV8+LZ5t1Rg0umU=">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</latexit><latexit sha1_base64="jYzScHcJllG/zV8+LZ5t1Rg0umU=">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</latexit><latexit sha1_base64="jYzScHcJllG/zV8+LZ5t1Rg0umU=">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</latexit>

- Linear independence:

↵1x1 + ↵2x2 + · · ·+ ↵kxk = 0 ) ↵i = 0, 8i<latexit sha1_base64="1cb+YYnCt14ghSXRVIWJBMZ7Ndo=">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</latexit><latexit sha1_base64="1cb+YYnCt14ghSXRVIWJBMZ7Ndo=">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</latexit><latexit sha1_base64="1cb+YYnCt14ghSXRVIWJBMZ7Ndo=">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</latexit><latexit sha1_base64="1cb+YYnCt14ghSXRVIWJBMZ7Ndo=">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</latexit>

Page 109: COMP 414/514: Optimization - Algorithms, Complexity and

Vectors- Span of a set of vectors:

span {x1, x2, . . . , xk} = {↵1x1 + ↵2x2 + · · ·+ ↵kxk | ↵i 2 R, i = [1, k]}<latexit sha1_base64="jYzScHcJllG/zV8+LZ5t1Rg0umU=">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</latexit><latexit sha1_base64="jYzScHcJllG/zV8+LZ5t1Rg0umU=">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</latexit><latexit sha1_base64="jYzScHcJllG/zV8+LZ5t1Rg0umU=">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</latexit><latexit sha1_base64="jYzScHcJllG/zV8+LZ5t1Rg0umU=">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</latexit>

- Linear independence:

↵1x1 + ↵2x2 + · · ·+ ↵kxk = 0 ) ↵i = 0, 8i<latexit sha1_base64="1cb+YYnCt14ghSXRVIWJBMZ7Ndo=">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</latexit><latexit sha1_base64="1cb+YYnCt14ghSXRVIWJBMZ7Ndo=">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</latexit><latexit sha1_base64="1cb+YYnCt14ghSXRVIWJBMZ7Ndo=">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</latexit><latexit sha1_base64="1cb+YYnCt14ghSXRVIWJBMZ7Ndo=">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</latexit>

- How does compare to , the vector dimension? k<latexit sha1_base64="FN4g9GFe+3ziaTgM0DMwdUeUKp0=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEqMeiF48t2A9oQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikreNUMWyxWMSqG1CNgktsGW4EdhOFNAoEdoLJ3dzvPKHSPJYPZpqgH9GR5CFn1FipORmUK27VXYCsEy8nFcjRGJS/+sOYpRFKwwTVuue5ifEzqgxnAmelfqoxoWxCR9izVNIItZ8tDp2RC6sMSRgrW9KQhfp7IqOR1tMosJ0RNWO96s3F/7xeasIbP+MySQ1KtlwUpoKYmMy/JkOukBkxtYQyxe2thI2poszYbEo2BG/15XXSvqp6btVrXlfqt3kcRTiDc7gED2pQh3toQAsYIDzDK7w5j86L8+58LFsLTj5zCn/gfP4A0oWM7w==</latexit><latexit sha1_base64="FN4g9GFe+3ziaTgM0DMwdUeUKp0=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEqMeiF48t2A9oQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikreNUMWyxWMSqG1CNgktsGW4EdhOFNAoEdoLJ3dzvPKHSPJYPZpqgH9GR5CFn1FipORmUK27VXYCsEy8nFcjRGJS/+sOYpRFKwwTVuue5ifEzqgxnAmelfqoxoWxCR9izVNIItZ8tDp2RC6sMSRgrW9KQhfp7IqOR1tMosJ0RNWO96s3F/7xeasIbP+MySQ1KtlwUpoKYmMy/JkOukBkxtYQyxe2thI2poszYbEo2BG/15XXSvqp6btVrXlfqt3kcRTiDc7gED2pQh3toQAsYIDzDK7w5j86L8+58LFsLTj5zCn/gfP4A0oWM7w==</latexit><latexit sha1_base64="FN4g9GFe+3ziaTgM0DMwdUeUKp0=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEqMeiF48t2A9oQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikreNUMWyxWMSqG1CNgktsGW4EdhOFNAoEdoLJ3dzvPKHSPJYPZpqgH9GR5CFn1FipORmUK27VXYCsEy8nFcjRGJS/+sOYpRFKwwTVuue5ifEzqgxnAmelfqoxoWxCR9izVNIItZ8tDp2RC6sMSRgrW9KQhfp7IqOR1tMosJ0RNWO96s3F/7xeasIbP+MySQ1KtlwUpoKYmMy/JkOukBkxtYQyxe2thI2poszYbEo2BG/15XXSvqp6btVrXlfqt3kcRTiDc7gED2pQh3toQAsYIDzDK7w5j86L8+58LFsLTj5zCn/gfP4A0oWM7w==</latexit><latexit sha1_base64="FN4g9GFe+3ziaTgM0DMwdUeUKp0=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEqMeiF48t2A9oQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikreNUMWyxWMSqG1CNgktsGW4EdhOFNAoEdoLJ3dzvPKHSPJYPZpqgH9GR5CFn1FipORmUK27VXYCsEy8nFcjRGJS/+sOYpRFKwwTVuue5ifEzqgxnAmelfqoxoWxCR9izVNIItZ8tDp2RC6sMSRgrW9KQhfp7IqOR1tMosJ0RNWO96s3F/7xeasIbP+MySQ1KtlwUpoKYmMy/JkOukBkxtYQyxe2thI2poszYbEo2BG/15XXSvqp6btVrXlfqt3kcRTiDc7gED2pQh3toQAsYIDzDK7w5j86L8+58LFsLTj5zCn/gfP4A0oWM7w==</latexit> p

<latexit sha1_base64="C50pkrGeTYlIc0VWLblfqsQv1p8=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEqMeiF48t2A9oQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikreNUMWyxWMSqG1CNgktsGW4EdhOFNAoEdoLJ3dzvPKHSPJYPZpqgH9GR5CFn1FipmQzKFbfqLkDWiZeTCuRoDMpf/WHM0gilYYJq3fPcxPgZVYYzgbNSP9WYUDahI+xZKmmE2s8Wh87IhVWGJIyVLWnIQv09kdFI62kU2M6ImrFe9ebif14vNeGNn3GZpAYlWy4KU0FMTOZfkyFXyIyYWkKZ4vZWwsZUUWZsNiUbgrf68jppX1U9t+o1ryv12zyOIpzBOVyCBzWowz00oAUMEJ7hFd6cR+fFeXc+lq0FJ585hT9wPn8A2hmM9A==</latexit><latexit sha1_base64="C50pkrGeTYlIc0VWLblfqsQv1p8=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEqMeiF48t2A9oQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikreNUMWyxWMSqG1CNgktsGW4EdhOFNAoEdoLJ3dzvPKHSPJYPZpqgH9GR5CFn1FipmQzKFbfqLkDWiZeTCuRoDMpf/WHM0gilYYJq3fPcxPgZVYYzgbNSP9WYUDahI+xZKmmE2s8Wh87IhVWGJIyVLWnIQv09kdFI62kU2M6ImrFe9ebif14vNeGNn3GZpAYlWy4KU0FMTOZfkyFXyIyYWkKZ4vZWwsZUUWZsNiUbgrf68jppX1U9t+o1ryv12zyOIpzBOVyCBzWowz00oAUMEJ7hFd6cR+fFeXc+lq0FJ585hT9wPn8A2hmM9A==</latexit><latexit sha1_base64="C50pkrGeTYlIc0VWLblfqsQv1p8=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEqMeiF48t2A9oQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikreNUMWyxWMSqG1CNgktsGW4EdhOFNAoEdoLJ3dzvPKHSPJYPZpqgH9GR5CFn1FipmQzKFbfqLkDWiZeTCuRoDMpf/WHM0gilYYJq3fPcxPgZVYYzgbNSP9WYUDahI+xZKmmE2s8Wh87IhVWGJIyVLWnIQv09kdFI62kU2M6ImrFe9ebif14vNeGNn3GZpAYlWy4KU0FMTOZfkyFXyIyYWkKZ4vZWwsZUUWZsNiUbgrf68jppX1U9t+o1ryv12zyOIpzBOVyCBzWowz00oAUMEJ7hFd6cR+fFeXc+lq0FJ585hT9wPn8A2hmM9A==</latexit><latexit sha1_base64="C50pkrGeTYlIc0VWLblfqsQv1p8=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEqMeiF48t2A9oQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikreNUMWyxWMSqG1CNgktsGW4EdhOFNAoEdoLJ3dzvPKHSPJYPZpqgH9GR5CFn1FipmQzKFbfqLkDWiZeTCuRoDMpf/WHM0gilYYJq3fPcxPgZVYYzgbNSP9WYUDahI+xZKmmE2s8Wh87IhVWGJIyVLWnIQv09kdFI62kU2M6ImrFe9ebif14vNeGNn3GZpAYlWy4KU0FMTOZfkyFXyIyYWkKZ4vZWwsZUUWZsNiUbgrf68jppX1U9t+o1ryv12zyOIpzBOVyCBzWowz00oAUMEJ7hFd6cR+fFeXc+lq0FJ585hT9wPn8A2hmM9A==</latexit>

Page 110: COMP 414/514: Optimization - Algorithms, Complexity and

Vectors- Span of a set of vectors:

span {x1, x2, . . . , xk} = {↵1x1 + ↵2x2 + · · ·+ ↵kxk | ↵i 2 R, i = [1, k]}<latexit sha1_base64="jYzScHcJllG/zV8+LZ5t1Rg0umU=">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</latexit><latexit sha1_base64="jYzScHcJllG/zV8+LZ5t1Rg0umU=">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</latexit><latexit sha1_base64="jYzScHcJllG/zV8+LZ5t1Rg0umU=">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</latexit><latexit sha1_base64="jYzScHcJllG/zV8+LZ5t1Rg0umU=">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</latexit>

- Linear independence:

↵1x1 + ↵2x2 + · · ·+ ↵kxk = 0 ) ↵i = 0, 8i<latexit sha1_base64="1cb+YYnCt14ghSXRVIWJBMZ7Ndo=">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</latexit><latexit sha1_base64="1cb+YYnCt14ghSXRVIWJBMZ7Ndo=">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</latexit><latexit sha1_base64="1cb+YYnCt14ghSXRVIWJBMZ7Ndo=">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</latexit><latexit sha1_base64="1cb+YYnCt14ghSXRVIWJBMZ7Ndo=">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</latexit>

- How does compare to , the vector dimension? k<latexit sha1_base64="FN4g9GFe+3ziaTgM0DMwdUeUKp0=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEqMeiF48t2A9oQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikreNUMWyxWMSqG1CNgktsGW4EdhOFNAoEdoLJ3dzvPKHSPJYPZpqgH9GR5CFn1FipORmUK27VXYCsEy8nFcjRGJS/+sOYpRFKwwTVuue5ifEzqgxnAmelfqoxoWxCR9izVNIItZ8tDp2RC6sMSRgrW9KQhfp7IqOR1tMosJ0RNWO96s3F/7xeasIbP+MySQ1KtlwUpoKYmMy/JkOukBkxtYQyxe2thI2poszYbEo2BG/15XXSvqp6btVrXlfqt3kcRTiDc7gED2pQh3toQAsYIDzDK7w5j86L8+58LFsLTj5zCn/gfP4A0oWM7w==</latexit><latexit sha1_base64="FN4g9GFe+3ziaTgM0DMwdUeUKp0=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEqMeiF48t2A9oQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikreNUMWyxWMSqG1CNgktsGW4EdhOFNAoEdoLJ3dzvPKHSPJYPZpqgH9GR5CFn1FipORmUK27VXYCsEy8nFcjRGJS/+sOYpRFKwwTVuue5ifEzqgxnAmelfqoxoWxCR9izVNIItZ8tDp2RC6sMSRgrW9KQhfp7IqOR1tMosJ0RNWO96s3F/7xeasIbP+MySQ1KtlwUpoKYmMy/JkOukBkxtYQyxe2thI2poszYbEo2BG/15XXSvqp6btVrXlfqt3kcRTiDc7gED2pQh3toQAsYIDzDK7w5j86L8+58LFsLTj5zCn/gfP4A0oWM7w==</latexit><latexit sha1_base64="FN4g9GFe+3ziaTgM0DMwdUeUKp0=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEqMeiF48t2A9oQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikreNUMWyxWMSqG1CNgktsGW4EdhOFNAoEdoLJ3dzvPKHSPJYPZpqgH9GR5CFn1FipORmUK27VXYCsEy8nFcjRGJS/+sOYpRFKwwTVuue5ifEzqgxnAmelfqoxoWxCR9izVNIItZ8tDp2RC6sMSRgrW9KQhfp7IqOR1tMosJ0RNWO96s3F/7xeasIbP+MySQ1KtlwUpoKYmMy/JkOukBkxtYQyxe2thI2poszYbEo2BG/15XXSvqp6btVrXlfqt3kcRTiDc7gED2pQh3toQAsYIDzDK7w5j86L8+58LFsLTj5zCn/gfP4A0oWM7w==</latexit><latexit sha1_base64="FN4g9GFe+3ziaTgM0DMwdUeUKp0=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEqMeiF48t2A9oQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikreNUMWyxWMSqG1CNgktsGW4EdhOFNAoEdoLJ3dzvPKHSPJYPZpqgH9GR5CFn1FipORmUK27VXYCsEy8nFcjRGJS/+sOYpRFKwwTVuue5ifEzqgxnAmelfqoxoWxCR9izVNIItZ8tDp2RC6sMSRgrW9KQhfp7IqOR1tMosJ0RNWO96s3F/7xeasIbP+MySQ1KtlwUpoKYmMy/JkOukBkxtYQyxe2thI2poszYbEo2BG/15XXSvqp6btVrXlfqt3kcRTiDc7gED2pQh3toQAsYIDzDK7w5j86L8+58LFsLTj5zCn/gfP4A0oWM7w==</latexit> p

<latexit sha1_base64="C50pkrGeTYlIc0VWLblfqsQv1p8=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEqMeiF48t2A9oQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikreNUMWyxWMSqG1CNgktsGW4EdhOFNAoEdoLJ3dzvPKHSPJYPZpqgH9GR5CFn1FipmQzKFbfqLkDWiZeTCuRoDMpf/WHM0gilYYJq3fPcxPgZVYYzgbNSP9WYUDahI+xZKmmE2s8Wh87IhVWGJIyVLWnIQv09kdFI62kU2M6ImrFe9ebif14vNeGNn3GZpAYlWy4KU0FMTOZfkyFXyIyYWkKZ4vZWwsZUUWZsNiUbgrf68jppX1U9t+o1ryv12zyOIpzBOVyCBzWowz00oAUMEJ7hFd6cR+fFeXc+lq0FJ585hT9wPn8A2hmM9A==</latexit><latexit sha1_base64="C50pkrGeTYlIc0VWLblfqsQv1p8=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEqMeiF48t2A9oQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikreNUMWyxWMSqG1CNgktsGW4EdhOFNAoEdoLJ3dzvPKHSPJYPZpqgH9GR5CFn1FipmQzKFbfqLkDWiZeTCuRoDMpf/WHM0gilYYJq3fPcxPgZVYYzgbNSP9WYUDahI+xZKmmE2s8Wh87IhVWGJIyVLWnIQv09kdFI62kU2M6ImrFe9ebif14vNeGNn3GZpAYlWy4KU0FMTOZfkyFXyIyYWkKZ4vZWwsZUUWZsNiUbgrf68jppX1U9t+o1ryv12zyOIpzBOVyCBzWowz00oAUMEJ7hFd6cR+fFeXc+lq0FJ585hT9wPn8A2hmM9A==</latexit><latexit sha1_base64="C50pkrGeTYlIc0VWLblfqsQv1p8=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEqMeiF48t2A9oQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikreNUMWyxWMSqG1CNgktsGW4EdhOFNAoEdoLJ3dzvPKHSPJYPZpqgH9GR5CFn1FipmQzKFbfqLkDWiZeTCuRoDMpf/WHM0gilYYJq3fPcxPgZVYYzgbNSP9WYUDahI+xZKmmE2s8Wh87IhVWGJIyVLWnIQv09kdFI62kU2M6ImrFe9ebif14vNeGNn3GZpAYlWy4KU0FMTOZfkyFXyIyYWkKZ4vZWwsZUUWZsNiUbgrf68jppX1U9t+o1ryv12zyOIpzBOVyCBzWowz00oAUMEJ7hFd6cR+fFeXc+lq0FJ585hT9wPn8A2hmM9A==</latexit><latexit sha1_base64="C50pkrGeTYlIc0VWLblfqsQv1p8=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEqMeiF48t2A9oQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikreNUMWyxWMSqG1CNgktsGW4EdhOFNAoEdoLJ3dzvPKHSPJYPZpqgH9GR5CFn1FipmQzKFbfqLkDWiZeTCuRoDMpf/WHM0gilYYJq3fPcxPgZVYYzgbNSP9WYUDahI+xZKmmE2s8Wh87IhVWGJIyVLWnIQv09kdFI62kU2M6ImrFe9ebif14vNeGNn3GZpAYlWy4KU0FMTOZfkyFXyIyYWkKZ4vZWwsZUUWZsNiUbgrf68jppX1U9t+o1ryv12zyOIpzBOVyCBzWowz00oAUMEJ7hFd6cR+fFeXc+lq0FJ585hT9wPn8A2hmM9A==</latexit>

- Inner product:

x>y = hx, yi =pX

i=1

xi · yi<latexit sha1_base64="hwf1qeWkkXHbH0wHYIgLioCPhZY=">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</latexit><latexit sha1_base64="hwf1qeWkkXHbH0wHYIgLioCPhZY=">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</latexit><latexit sha1_base64="hwf1qeWkkXHbH0wHYIgLioCPhZY=">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</latexit><latexit sha1_base64="hwf1qeWkkXHbH0wHYIgLioCPhZY=">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</latexit>

Page 111: COMP 414/514: Optimization - Algorithms, Complexity and

Vectors- Inner production interpretation:

hx, yi = kxk · kyk · cos ✓<latexit sha1_base64="Erkef27IuCLz13CGlmdw2LKnN48=">AAACL3icbZDLSgMxFIYz3q23qks3wSK4EJkRQTeCKIhLBatCU0omPdMGM8mQnJEOrW/kxldxI6KIW9/C9AJeDwQ+vnNCcv44U9JhGD4HY+MTk1PTM7OlufmFxaXy8sqlM7kVUBVGGXsdcwdKaqiiRAXXmQWexgqu4pvjfv/qFqyTRl9gkUE95S0tEyk4etUonzAFCVKmuG4poJ0tWlBmZavtnR26A8p6HdajTDSNt73ii4VxDNuAvFGuhNvhoOhfiEZQIaM6a5QfWdOIPAWNQnHnalGYYb3LLUqh4K7EcgcZFze8BTWPmqfg6t3Bvnd0w5smTYz1RyMd2O83ujx1rkhjP5lybLvfvb78r1fLMdmvd6XOcgQthg8luaJoaD882pQWBKrCAxdW+r9S0eaWC/QRl3wI0e+V/8Llznbk+Xy3cng0imOGrJF1skkiskcOySk5I1UiyD15JC/kNXgInoK34H04OhaM7qySHxV8fAJp7Ki2</latexit><latexit sha1_base64="Erkef27IuCLz13CGlmdw2LKnN48=">AAACL3icbZDLSgMxFIYz3q23qks3wSK4EJkRQTeCKIhLBatCU0omPdMGM8mQnJEOrW/kxldxI6KIW9/C9AJeDwQ+vnNCcv44U9JhGD4HY+MTk1PTM7OlufmFxaXy8sqlM7kVUBVGGXsdcwdKaqiiRAXXmQWexgqu4pvjfv/qFqyTRl9gkUE95S0tEyk4etUonzAFCVKmuG4poJ0tWlBmZavtnR26A8p6HdajTDSNt73ii4VxDNuAvFGuhNvhoOhfiEZQIaM6a5QfWdOIPAWNQnHnalGYYb3LLUqh4K7EcgcZFze8BTWPmqfg6t3Bvnd0w5smTYz1RyMd2O83ujx1rkhjP5lybLvfvb78r1fLMdmvd6XOcgQthg8luaJoaD882pQWBKrCAxdW+r9S0eaWC/QRl3wI0e+V/8Llznbk+Xy3cng0imOGrJF1skkiskcOySk5I1UiyD15JC/kNXgInoK34H04OhaM7qySHxV8fAJp7Ki2</latexit><latexit sha1_base64="Erkef27IuCLz13CGlmdw2LKnN48=">AAACL3icbZDLSgMxFIYz3q23qks3wSK4EJkRQTeCKIhLBatCU0omPdMGM8mQnJEOrW/kxldxI6KIW9/C9AJeDwQ+vnNCcv44U9JhGD4HY+MTk1PTM7OlufmFxaXy8sqlM7kVUBVGGXsdcwdKaqiiRAXXmQWexgqu4pvjfv/qFqyTRl9gkUE95S0tEyk4etUonzAFCVKmuG4poJ0tWlBmZavtnR26A8p6HdajTDSNt73ii4VxDNuAvFGuhNvhoOhfiEZQIaM6a5QfWdOIPAWNQnHnalGYYb3LLUqh4K7EcgcZFze8BTWPmqfg6t3Bvnd0w5smTYz1RyMd2O83ujx1rkhjP5lybLvfvb78r1fLMdmvd6XOcgQthg8luaJoaD882pQWBKrCAxdW+r9S0eaWC/QRl3wI0e+V/8Llznbk+Xy3cng0imOGrJF1skkiskcOySk5I1UiyD15JC/kNXgInoK34H04OhaM7qySHxV8fAJp7Ki2</latexit><latexit sha1_base64="Erkef27IuCLz13CGlmdw2LKnN48=">AAACL3icbZDLSgMxFIYz3q23qks3wSK4EJkRQTeCKIhLBatCU0omPdMGM8mQnJEOrW/kxldxI6KIW9/C9AJeDwQ+vnNCcv44U9JhGD4HY+MTk1PTM7OlufmFxaXy8sqlM7kVUBVGGXsdcwdKaqiiRAXXmQWexgqu4pvjfv/qFqyTRl9gkUE95S0tEyk4etUonzAFCVKmuG4poJ0tWlBmZavtnR26A8p6HdajTDSNt73ii4VxDNuAvFGuhNvhoOhfiEZQIaM6a5QfWdOIPAWNQnHnalGYYb3LLUqh4K7EcgcZFze8BTWPmqfg6t3Bvnd0w5smTYz1RyMd2O83ujx1rkhjP5lybLvfvb78r1fLMdmvd6XOcgQthg8luaJoaD882pQWBKrCAxdW+r9S0eaWC/QRl3wI0e+V/8Llznbk+Xy3cng0imOGrJF1skkiskcOySk5I1UiyD15JC/kNXgInoK34H04OhaM7qySHxV8fAJp7Ki2</latexit>

Page 112: COMP 414/514: Optimization - Algorithms, Complexity and

Vectors- Inner production interpretation:

x<latexit sha1_base64="IArQpDG4Gw7Ax+5Wri9CZWKD4Bo=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfRY9OKxBfsBbSib7aRdu9mE3Y1YQn+BFw+KePUnefPfuG1z0NYXFh7emWFn3iARXBvX/XYKa+sbm1vF7dLO7t7+QfnwqKXjVDFssljEqhNQjYJLbBpuBHYShTQKBLaD8e2s3n5EpXks780kQT+iQ8lDzqixVuOpX664VXcusgpeDhXIVe+Xv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtShphNrP5otOyZl1BiSMlX3SkLn7eyKjkdaTKLCdETUjvVybmf/VuqkJr/2MyyQ1KNniozAVxMRkdjUZcIXMiIkFyhS3uxI2oooyY7Mp2RC85ZNXoXVR9Sw3Liu1mzyOIpzAKZyDB1dQgzuoQxMYIDzDK7w5D86L8+58LFoLTj5zDH/kfP4A5juM/A==</latexit><latexit sha1_base64="IArQpDG4Gw7Ax+5Wri9CZWKD4Bo=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfRY9OKxBfsBbSib7aRdu9mE3Y1YQn+BFw+KePUnefPfuG1z0NYXFh7emWFn3iARXBvX/XYKa+sbm1vF7dLO7t7+QfnwqKXjVDFssljEqhNQjYJLbBpuBHYShTQKBLaD8e2s3n5EpXks780kQT+iQ8lDzqixVuOpX664VXcusgpeDhXIVe+Xv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtShphNrP5otOyZl1BiSMlX3SkLn7eyKjkdaTKLCdETUjvVybmf/VuqkJr/2MyyQ1KNniozAVxMRkdjUZcIXMiIkFyhS3uxI2oooyY7Mp2RC85ZNXoXVR9Sw3Liu1mzyOIpzAKZyDB1dQgzuoQxMYIDzDK7w5D86L8+58LFoLTj5zDH/kfP4A5juM/A==</latexit><latexit sha1_base64="IArQpDG4Gw7Ax+5Wri9CZWKD4Bo=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfRY9OKxBfsBbSib7aRdu9mE3Y1YQn+BFw+KePUnefPfuG1z0NYXFh7emWFn3iARXBvX/XYKa+sbm1vF7dLO7t7+QfnwqKXjVDFssljEqhNQjYJLbBpuBHYShTQKBLaD8e2s3n5EpXks780kQT+iQ8lDzqixVuOpX664VXcusgpeDhXIVe+Xv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtShphNrP5otOyZl1BiSMlX3SkLn7eyKjkdaTKLCdETUjvVybmf/VuqkJr/2MyyQ1KNniozAVxMRkdjUZcIXMiIkFyhS3uxI2oooyY7Mp2RC85ZNXoXVR9Sw3Liu1mzyOIpzAKZyDB1dQgzuoQxMYIDzDK7w5D86L8+58LFoLTj5zDH/kfP4A5juM/A==</latexit><latexit sha1_base64="IArQpDG4Gw7Ax+5Wri9CZWKD4Bo=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfRY9OKxBfsBbSib7aRdu9mE3Y1YQn+BFw+KePUnefPfuG1z0NYXFh7emWFn3iARXBvX/XYKa+sbm1vF7dLO7t7+QfnwqKXjVDFssljEqhNQjYJLbBpuBHYShTQKBLaD8e2s3n5EpXks780kQT+iQ8lDzqixVuOpX664VXcusgpeDhXIVe+Xv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtShphNrP5otOyZl1BiSMlX3SkLn7eyKjkdaTKLCdETUjvVybmf/VuqkJr/2MyyQ1KNniozAVxMRkdjUZcIXMiIkFyhS3uxI2oooyY7Mp2RC85ZNXoXVR9Sw3Liu1mzyOIpzAKZyDB1dQgzuoQxMYIDzDK7w5D86L8+58LFoLTj5zDH/kfP4A5juM/A==</latexit>

y<latexit sha1_base64="kBWpEtbVW3xlz0t9EJbdPDfcZn0=">AAAB6HicbZBNS8NAEIYnftb6VfXoZbEInkoigh6LXjy2YD+gDWWznbRrN5uwuxFC6C/w4kERr/4kb/4bt20O2vrCwsM7M+zMGySCa+O6387a+sbm1nZpp7y7t39wWDk6bus4VQxbLBax6gZUo+ASW4Ybgd1EIY0CgZ1gcjerd55QaR7LB5Ml6Ed0JHnIGTXWamaDStWtuXORVfAKqEKhxqDy1R/GLI1QGiao1j3PTYyfU2U4Ezgt91ONCWUTOsKeRUkj1H4+X3RKzq0zJGGs7JOGzN3fEzmNtM6iwHZG1Iz1cm1m/lfrpSa88XMuk9SgZIuPwlQQE5PZ1WTIFTIjMguUKW53JWxMFWXGZlO2IXjLJ69C+7LmWW5eVeu3RRwlOIUzuAAPrqEO99CAFjBAeIZXeHMenRfn3flYtK45xcwJ/JHz+QPnv4z9</latexit><latexit sha1_base64="kBWpEtbVW3xlz0t9EJbdPDfcZn0=">AAAB6HicbZBNS8NAEIYnftb6VfXoZbEInkoigh6LXjy2YD+gDWWznbRrN5uwuxFC6C/w4kERr/4kb/4bt20O2vrCwsM7M+zMGySCa+O6387a+sbm1nZpp7y7t39wWDk6bus4VQxbLBax6gZUo+ASW4Ybgd1EIY0CgZ1gcjerd55QaR7LB5Ml6Ed0JHnIGTXWamaDStWtuXORVfAKqEKhxqDy1R/GLI1QGiao1j3PTYyfU2U4Ezgt91ONCWUTOsKeRUkj1H4+X3RKzq0zJGGs7JOGzN3fEzmNtM6iwHZG1Iz1cm1m/lfrpSa88XMuk9SgZIuPwlQQE5PZ1WTIFTIjMguUKW53JWxMFWXGZlO2IXjLJ69C+7LmWW5eVeu3RRwlOIUzuAAPrqEO99CAFjBAeIZXeHMenRfn3flYtK45xcwJ/JHz+QPnv4z9</latexit><latexit sha1_base64="kBWpEtbVW3xlz0t9EJbdPDfcZn0=">AAAB6HicbZBNS8NAEIYnftb6VfXoZbEInkoigh6LXjy2YD+gDWWznbRrN5uwuxFC6C/w4kERr/4kb/4bt20O2vrCwsM7M+zMGySCa+O6387a+sbm1nZpp7y7t39wWDk6bus4VQxbLBax6gZUo+ASW4Ybgd1EIY0CgZ1gcjerd55QaR7LB5Ml6Ed0JHnIGTXWamaDStWtuXORVfAKqEKhxqDy1R/GLI1QGiao1j3PTYyfU2U4Ezgt91ONCWUTOsKeRUkj1H4+X3RKzq0zJGGs7JOGzN3fEzmNtM6iwHZG1Iz1cm1m/lfrpSa88XMuk9SgZIuPwlQQE5PZ1WTIFTIjMguUKW53JWxMFWXGZlO2IXjLJ69C+7LmWW5eVeu3RRwlOIUzuAAPrqEO99CAFjBAeIZXeHMenRfn3flYtK45xcwJ/JHz+QPnv4z9</latexit><latexit sha1_base64="kBWpEtbVW3xlz0t9EJbdPDfcZn0=">AAAB6HicbZBNS8NAEIYnftb6VfXoZbEInkoigh6LXjy2YD+gDWWznbRrN5uwuxFC6C/w4kERr/4kb/4bt20O2vrCwsM7M+zMGySCa+O6387a+sbm1nZpp7y7t39wWDk6bus4VQxbLBax6gZUo+ASW4Ybgd1EIY0CgZ1gcjerd55QaR7LB5Ml6Ed0JHnIGTXWamaDStWtuXORVfAKqEKhxqDy1R/GLI1QGiao1j3PTYyfU2U4Ezgt91ONCWUTOsKeRUkj1H4+X3RKzq0zJGGs7JOGzN3fEzmNtM6iwHZG1Iz1cm1m/lfrpSa88XMuk9SgZIuPwlQQE5PZ1WTIFTIjMguUKW53JWxMFWXGZlO2IXjLJ69C+7LmWW5eVeu3RRwlOIUzuAAPrqEO99CAFjBAeIZXeHMenRfn3flYtK45xcwJ/JHz+QPnv4z9</latexit>

hx, yi = kxk · kyk · cos ✓<latexit sha1_base64="Erkef27IuCLz13CGlmdw2LKnN48=">AAACL3icbZDLSgMxFIYz3q23qks3wSK4EJkRQTeCKIhLBatCU0omPdMGM8mQnJEOrW/kxldxI6KIW9/C9AJeDwQ+vnNCcv44U9JhGD4HY+MTk1PTM7OlufmFxaXy8sqlM7kVUBVGGXsdcwdKaqiiRAXXmQWexgqu4pvjfv/qFqyTRl9gkUE95S0tEyk4etUonzAFCVKmuG4poJ0tWlBmZavtnR26A8p6HdajTDSNt73ii4VxDNuAvFGuhNvhoOhfiEZQIaM6a5QfWdOIPAWNQnHnalGYYb3LLUqh4K7EcgcZFze8BTWPmqfg6t3Bvnd0w5smTYz1RyMd2O83ujx1rkhjP5lybLvfvb78r1fLMdmvd6XOcgQthg8luaJoaD882pQWBKrCAxdW+r9S0eaWC/QRl3wI0e+V/8Llznbk+Xy3cng0imOGrJF1skkiskcOySk5I1UiyD15JC/kNXgInoK34H04OhaM7qySHxV8fAJp7Ki2</latexit><latexit sha1_base64="Erkef27IuCLz13CGlmdw2LKnN48=">AAACL3icbZDLSgMxFIYz3q23qks3wSK4EJkRQTeCKIhLBatCU0omPdMGM8mQnJEOrW/kxldxI6KIW9/C9AJeDwQ+vnNCcv44U9JhGD4HY+MTk1PTM7OlufmFxaXy8sqlM7kVUBVGGXsdcwdKaqiiRAXXmQWexgqu4pvjfv/qFqyTRl9gkUE95S0tEyk4etUonzAFCVKmuG4poJ0tWlBmZavtnR26A8p6HdajTDSNt73ii4VxDNuAvFGuhNvhoOhfiEZQIaM6a5QfWdOIPAWNQnHnalGYYb3LLUqh4K7EcgcZFze8BTWPmqfg6t3Bvnd0w5smTYz1RyMd2O83ujx1rkhjP5lybLvfvb78r1fLMdmvd6XOcgQthg8luaJoaD882pQWBKrCAxdW+r9S0eaWC/QRl3wI0e+V/8Llznbk+Xy3cng0imOGrJF1skkiskcOySk5I1UiyD15JC/kNXgInoK34H04OhaM7qySHxV8fAJp7Ki2</latexit><latexit sha1_base64="Erkef27IuCLz13CGlmdw2LKnN48=">AAACL3icbZDLSgMxFIYz3q23qks3wSK4EJkRQTeCKIhLBatCU0omPdMGM8mQnJEOrW/kxldxI6KIW9/C9AJeDwQ+vnNCcv44U9JhGD4HY+MTk1PTM7OlufmFxaXy8sqlM7kVUBVGGXsdcwdKaqiiRAXXmQWexgqu4pvjfv/qFqyTRl9gkUE95S0tEyk4etUonzAFCVKmuG4poJ0tWlBmZavtnR26A8p6HdajTDSNt73ii4VxDNuAvFGuhNvhoOhfiEZQIaM6a5QfWdOIPAWNQnHnalGYYb3LLUqh4K7EcgcZFze8BTWPmqfg6t3Bvnd0w5smTYz1RyMd2O83ujx1rkhjP5lybLvfvb78r1fLMdmvd6XOcgQthg8luaJoaD882pQWBKrCAxdW+r9S0eaWC/QRl3wI0e+V/8Llznbk+Xy3cng0imOGrJF1skkiskcOySk5I1UiyD15JC/kNXgInoK34H04OhaM7qySHxV8fAJp7Ki2</latexit><latexit sha1_base64="Erkef27IuCLz13CGlmdw2LKnN48=">AAACL3icbZDLSgMxFIYz3q23qks3wSK4EJkRQTeCKIhLBatCU0omPdMGM8mQnJEOrW/kxldxI6KIW9/C9AJeDwQ+vnNCcv44U9JhGD4HY+MTk1PTM7OlufmFxaXy8sqlM7kVUBVGGXsdcwdKaqiiRAXXmQWexgqu4pvjfv/qFqyTRl9gkUE95S0tEyk4etUonzAFCVKmuG4poJ0tWlBmZavtnR26A8p6HdajTDSNt73ii4VxDNuAvFGuhNvhoOhfiEZQIaM6a5QfWdOIPAWNQnHnalGYYb3LLUqh4K7EcgcZFze8BTWPmqfg6t3Bvnd0w5smTYz1RyMd2O83ujx1rkhjP5lybLvfvb78r1fLMdmvd6XOcgQthg8luaJoaD882pQWBKrCAxdW+r9S0eaWC/QRl3wI0e+V/8Llznbk+Xy3cng0imOGrJF1skkiskcOySk5I1UiyD15JC/kNXgInoK34H04OhaM7qySHxV8fAJp7Ki2</latexit>

Page 113: COMP 414/514: Optimization - Algorithms, Complexity and

Vectors- Inner production interpretation:

x<latexit sha1_base64="IArQpDG4Gw7Ax+5Wri9CZWKD4Bo=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfRY9OKxBfsBbSib7aRdu9mE3Y1YQn+BFw+KePUnefPfuG1z0NYXFh7emWFn3iARXBvX/XYKa+sbm1vF7dLO7t7+QfnwqKXjVDFssljEqhNQjYJLbBpuBHYShTQKBLaD8e2s3n5EpXks780kQT+iQ8lDzqixVuOpX664VXcusgpeDhXIVe+Xv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtShphNrP5otOyZl1BiSMlX3SkLn7eyKjkdaTKLCdETUjvVybmf/VuqkJr/2MyyQ1KNniozAVxMRkdjUZcIXMiIkFyhS3uxI2oooyY7Mp2RC85ZNXoXVR9Sw3Liu1mzyOIpzAKZyDB1dQgzuoQxMYIDzDK7w5D86L8+58LFoLTj5zDH/kfP4A5juM/A==</latexit><latexit sha1_base64="IArQpDG4Gw7Ax+5Wri9CZWKD4Bo=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfRY9OKxBfsBbSib7aRdu9mE3Y1YQn+BFw+KePUnefPfuG1z0NYXFh7emWFn3iARXBvX/XYKa+sbm1vF7dLO7t7+QfnwqKXjVDFssljEqhNQjYJLbBpuBHYShTQKBLaD8e2s3n5EpXks780kQT+iQ8lDzqixVuOpX664VXcusgpeDhXIVe+Xv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtShphNrP5otOyZl1BiSMlX3SkLn7eyKjkdaTKLCdETUjvVybmf/VuqkJr/2MyyQ1KNniozAVxMRkdjUZcIXMiIkFyhS3uxI2oooyY7Mp2RC85ZNXoXVR9Sw3Liu1mzyOIpzAKZyDB1dQgzuoQxMYIDzDK7w5D86L8+58LFoLTj5zDH/kfP4A5juM/A==</latexit><latexit sha1_base64="IArQpDG4Gw7Ax+5Wri9CZWKD4Bo=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfRY9OKxBfsBbSib7aRdu9mE3Y1YQn+BFw+KePUnefPfuG1z0NYXFh7emWFn3iARXBvX/XYKa+sbm1vF7dLO7t7+QfnwqKXjVDFssljEqhNQjYJLbBpuBHYShTQKBLaD8e2s3n5EpXks780kQT+iQ8lDzqixVuOpX664VXcusgpeDhXIVe+Xv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtShphNrP5otOyZl1BiSMlX3SkLn7eyKjkdaTKLCdETUjvVybmf/VuqkJr/2MyyQ1KNniozAVxMRkdjUZcIXMiIkFyhS3uxI2oooyY7Mp2RC85ZNXoXVR9Sw3Liu1mzyOIpzAKZyDB1dQgzuoQxMYIDzDK7w5D86L8+58LFoLTj5zDH/kfP4A5juM/A==</latexit><latexit sha1_base64="IArQpDG4Gw7Ax+5Wri9CZWKD4Bo=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfRY9OKxBfsBbSib7aRdu9mE3Y1YQn+BFw+KePUnefPfuG1z0NYXFh7emWFn3iARXBvX/XYKa+sbm1vF7dLO7t7+QfnwqKXjVDFssljEqhNQjYJLbBpuBHYShTQKBLaD8e2s3n5EpXks780kQT+iQ8lDzqixVuOpX664VXcusgpeDhXIVe+Xv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtShphNrP5otOyZl1BiSMlX3SkLn7eyKjkdaTKLCdETUjvVybmf/VuqkJr/2MyyQ1KNniozAVxMRkdjUZcIXMiIkFyhS3uxI2oooyY7Mp2RC85ZNXoXVR9Sw3Liu1mzyOIpzAKZyDB1dQgzuoQxMYIDzDK7w5D86L8+58LFoLTj5zDH/kfP4A5juM/A==</latexit>

y<latexit sha1_base64="kBWpEtbVW3xlz0t9EJbdPDfcZn0=">AAAB6HicbZBNS8NAEIYnftb6VfXoZbEInkoigh6LXjy2YD+gDWWznbRrN5uwuxFC6C/w4kERr/4kb/4bt20O2vrCwsM7M+zMGySCa+O6387a+sbm1nZpp7y7t39wWDk6bus4VQxbLBax6gZUo+ASW4Ybgd1EIY0CgZ1gcjerd55QaR7LB5Ml6Ed0JHnIGTXWamaDStWtuXORVfAKqEKhxqDy1R/GLI1QGiao1j3PTYyfU2U4Ezgt91ONCWUTOsKeRUkj1H4+X3RKzq0zJGGs7JOGzN3fEzmNtM6iwHZG1Iz1cm1m/lfrpSa88XMuk9SgZIuPwlQQE5PZ1WTIFTIjMguUKW53JWxMFWXGZlO2IXjLJ69C+7LmWW5eVeu3RRwlOIUzuAAPrqEO99CAFjBAeIZXeHMenRfn3flYtK45xcwJ/JHz+QPnv4z9</latexit><latexit sha1_base64="kBWpEtbVW3xlz0t9EJbdPDfcZn0=">AAAB6HicbZBNS8NAEIYnftb6VfXoZbEInkoigh6LXjy2YD+gDWWznbRrN5uwuxFC6C/w4kERr/4kb/4bt20O2vrCwsM7M+zMGySCa+O6387a+sbm1nZpp7y7t39wWDk6bus4VQxbLBax6gZUo+ASW4Ybgd1EIY0CgZ1gcjerd55QaR7LB5Ml6Ed0JHnIGTXWamaDStWtuXORVfAKqEKhxqDy1R/GLI1QGiao1j3PTYyfU2U4Ezgt91ONCWUTOsKeRUkj1H4+X3RKzq0zJGGs7JOGzN3fEzmNtM6iwHZG1Iz1cm1m/lfrpSa88XMuk9SgZIuPwlQQE5PZ1WTIFTIjMguUKW53JWxMFWXGZlO2IXjLJ69C+7LmWW5eVeu3RRwlOIUzuAAPrqEO99CAFjBAeIZXeHMenRfn3flYtK45xcwJ/JHz+QPnv4z9</latexit><latexit sha1_base64="kBWpEtbVW3xlz0t9EJbdPDfcZn0=">AAAB6HicbZBNS8NAEIYnftb6VfXoZbEInkoigh6LXjy2YD+gDWWznbRrN5uwuxFC6C/w4kERr/4kb/4bt20O2vrCwsM7M+zMGySCa+O6387a+sbm1nZpp7y7t39wWDk6bus4VQxbLBax6gZUo+ASW4Ybgd1EIY0CgZ1gcjerd55QaR7LB5Ml6Ed0JHnIGTXWamaDStWtuXORVfAKqEKhxqDy1R/GLI1QGiao1j3PTYyfU2U4Ezgt91ONCWUTOsKeRUkj1H4+X3RKzq0zJGGs7JOGzN3fEzmNtM6iwHZG1Iz1cm1m/lfrpSa88XMuk9SgZIuPwlQQE5PZ1WTIFTIjMguUKW53JWxMFWXGZlO2IXjLJ69C+7LmWW5eVeu3RRwlOIUzuAAPrqEO99CAFjBAeIZXeHMenRfn3flYtK45xcwJ/JHz+QPnv4z9</latexit><latexit sha1_base64="kBWpEtbVW3xlz0t9EJbdPDfcZn0=">AAAB6HicbZBNS8NAEIYnftb6VfXoZbEInkoigh6LXjy2YD+gDWWznbRrN5uwuxFC6C/w4kERr/4kb/4bt20O2vrCwsM7M+zMGySCa+O6387a+sbm1nZpp7y7t39wWDk6bus4VQxbLBax6gZUo+ASW4Ybgd1EIY0CgZ1gcjerd55QaR7LB5Ml6Ed0JHnIGTXWamaDStWtuXORVfAKqEKhxqDy1R/GLI1QGiao1j3PTYyfU2U4Ezgt91ONCWUTOsKeRUkj1H4+X3RKzq0zJGGs7JOGzN3fEzmNtM6iwHZG1Iz1cm1m/lfrpSa88XMuk9SgZIuPwlQQE5PZ1WTIFTIjMguUKW53JWxMFWXGZlO2IXjLJ69C+7LmWW5eVeu3RRwlOIUzuAAPrqEO99CAFjBAeIZXeHMenRfn3flYtK45xcwJ/JHz+QPnv4z9</latexit>✓

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Page 114: COMP 414/514: Optimization - Algorithms, Complexity and

Vectors- Inner production interpretation:

x<latexit sha1_base64="IArQpDG4Gw7Ax+5Wri9CZWKD4Bo=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfRY9OKxBfsBbSib7aRdu9mE3Y1YQn+BFw+KePUnefPfuG1z0NYXFh7emWFn3iARXBvX/XYKa+sbm1vF7dLO7t7+QfnwqKXjVDFssljEqhNQjYJLbBpuBHYShTQKBLaD8e2s3n5EpXks780kQT+iQ8lDzqixVuOpX664VXcusgpeDhXIVe+Xv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtShphNrP5otOyZl1BiSMlX3SkLn7eyKjkdaTKLCdETUjvVybmf/VuqkJr/2MyyQ1KNniozAVxMRkdjUZcIXMiIkFyhS3uxI2oooyY7Mp2RC85ZNXoXVR9Sw3Liu1mzyOIpzAKZyDB1dQgzuoQxMYIDzDK7w5D86L8+58LFoLTj5zDH/kfP4A5juM/A==</latexit><latexit sha1_base64="IArQpDG4Gw7Ax+5Wri9CZWKD4Bo=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfRY9OKxBfsBbSib7aRdu9mE3Y1YQn+BFw+KePUnefPfuG1z0NYXFh7emWFn3iARXBvX/XYKa+sbm1vF7dLO7t7+QfnwqKXjVDFssljEqhNQjYJLbBpuBHYShTQKBLaD8e2s3n5EpXks780kQT+iQ8lDzqixVuOpX664VXcusgpeDhXIVe+Xv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtShphNrP5otOyZl1BiSMlX3SkLn7eyKjkdaTKLCdETUjvVybmf/VuqkJr/2MyyQ1KNniozAVxMRkdjUZcIXMiIkFyhS3uxI2oooyY7Mp2RC85ZNXoXVR9Sw3Liu1mzyOIpzAKZyDB1dQgzuoQxMYIDzDK7w5D86L8+58LFoLTj5zDH/kfP4A5juM/A==</latexit><latexit sha1_base64="IArQpDG4Gw7Ax+5Wri9CZWKD4Bo=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfRY9OKxBfsBbSib7aRdu9mE3Y1YQn+BFw+KePUnefPfuG1z0NYXFh7emWFn3iARXBvX/XYKa+sbm1vF7dLO7t7+QfnwqKXjVDFssljEqhNQjYJLbBpuBHYShTQKBLaD8e2s3n5EpXks780kQT+iQ8lDzqixVuOpX664VXcusgpeDhXIVe+Xv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtShphNrP5otOyZl1BiSMlX3SkLn7eyKjkdaTKLCdETUjvVybmf/VuqkJr/2MyyQ1KNniozAVxMRkdjUZcIXMiIkFyhS3uxI2oooyY7Mp2RC85ZNXoXVR9Sw3Liu1mzyOIpzAKZyDB1dQgzuoQxMYIDzDK7w5D86L8+58LFoLTj5zDH/kfP4A5juM/A==</latexit><latexit sha1_base64="IArQpDG4Gw7Ax+5Wri9CZWKD4Bo=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfRY9OKxBfsBbSib7aRdu9mE3Y1YQn+BFw+KePUnefPfuG1z0NYXFh7emWFn3iARXBvX/XYKa+sbm1vF7dLO7t7+QfnwqKXjVDFssljEqhNQjYJLbBpuBHYShTQKBLaD8e2s3n5EpXks780kQT+iQ8lDzqixVuOpX664VXcusgpeDhXIVe+Xv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtShphNrP5otOyZl1BiSMlX3SkLn7eyKjkdaTKLCdETUjvVybmf/VuqkJr/2MyyQ1KNniozAVxMRkdjUZcIXMiIkFyhS3uxI2oooyY7Mp2RC85ZNXoXVR9Sw3Liu1mzyOIpzAKZyDB1dQgzuoQxMYIDzDK7w5D86L8+58LFoLTj5zDH/kfP4A5juM/A==</latexit>

y<latexit sha1_base64="kBWpEtbVW3xlz0t9EJbdPDfcZn0=">AAAB6HicbZBNS8NAEIYnftb6VfXoZbEInkoigh6LXjy2YD+gDWWznbRrN5uwuxFC6C/w4kERr/4kb/4bt20O2vrCwsM7M+zMGySCa+O6387a+sbm1nZpp7y7t39wWDk6bus4VQxbLBax6gZUo+ASW4Ybgd1EIY0CgZ1gcjerd55QaR7LB5Ml6Ed0JHnIGTXWamaDStWtuXORVfAKqEKhxqDy1R/GLI1QGiao1j3PTYyfU2U4Ezgt91ONCWUTOsKeRUkj1H4+X3RKzq0zJGGs7JOGzN3fEzmNtM6iwHZG1Iz1cm1m/lfrpSa88XMuk9SgZIuPwlQQE5PZ1WTIFTIjMguUKW53JWxMFWXGZlO2IXjLJ69C+7LmWW5eVeu3RRwlOIUzuAAPrqEO99CAFjBAeIZXeHMenRfn3flYtK45xcwJ/JHz+QPnv4z9</latexit><latexit sha1_base64="kBWpEtbVW3xlz0t9EJbdPDfcZn0=">AAAB6HicbZBNS8NAEIYnftb6VfXoZbEInkoigh6LXjy2YD+gDWWznbRrN5uwuxFC6C/w4kERr/4kb/4bt20O2vrCwsM7M+zMGySCa+O6387a+sbm1nZpp7y7t39wWDk6bus4VQxbLBax6gZUo+ASW4Ybgd1EIY0CgZ1gcjerd55QaR7LB5Ml6Ed0JHnIGTXWamaDStWtuXORVfAKqEKhxqDy1R/GLI1QGiao1j3PTYyfU2U4Ezgt91ONCWUTOsKeRUkj1H4+X3RKzq0zJGGs7JOGzN3fEzmNtM6iwHZG1Iz1cm1m/lfrpSa88XMuk9SgZIuPwlQQE5PZ1WTIFTIjMguUKW53JWxMFWXGZlO2IXjLJ69C+7LmWW5eVeu3RRwlOIUzuAAPrqEO99CAFjBAeIZXeHMenRfn3flYtK45xcwJ/JHz+QPnv4z9</latexit><latexit sha1_base64="kBWpEtbVW3xlz0t9EJbdPDfcZn0=">AAAB6HicbZBNS8NAEIYnftb6VfXoZbEInkoigh6LXjy2YD+gDWWznbRrN5uwuxFC6C/w4kERr/4kb/4bt20O2vrCwsM7M+zMGySCa+O6387a+sbm1nZpp7y7t39wWDk6bus4VQxbLBax6gZUo+ASW4Ybgd1EIY0CgZ1gcjerd55QaR7LB5Ml6Ed0JHnIGTXWamaDStWtuXORVfAKqEKhxqDy1R/GLI1QGiao1j3PTYyfU2U4Ezgt91ONCWUTOsKeRUkj1H4+X3RKzq0zJGGs7JOGzN3fEzmNtM6iwHZG1Iz1cm1m/lfrpSa88XMuk9SgZIuPwlQQE5PZ1WTIFTIjMguUKW53JWxMFWXGZlO2IXjLJ69C+7LmWW5eVeu3RRwlOIUzuAAPrqEO99CAFjBAeIZXeHMenRfn3flYtK45xcwJ/JHz+QPnv4z9</latexit><latexit sha1_base64="kBWpEtbVW3xlz0t9EJbdPDfcZn0=">AAAB6HicbZBNS8NAEIYnftb6VfXoZbEInkoigh6LXjy2YD+gDWWznbRrN5uwuxFC6C/w4kERr/4kb/4bt20O2vrCwsM7M+zMGySCa+O6387a+sbm1nZpp7y7t39wWDk6bus4VQxbLBax6gZUo+ASW4Ybgd1EIY0CgZ1gcjerd55QaR7LB5Ml6Ed0JHnIGTXWamaDStWtuXORVfAKqEKhxqDy1R/GLI1QGiao1j3PTYyfU2U4Ezgt91ONCWUTOsKeRUkj1H4+X3RKzq0zJGGs7JOGzN3fEzmNtM6iwHZG1Iz1cm1m/lfrpSa88XMuk9SgZIuPwlQQE5PZ1WTIFTIjMguUKW53JWxMFWXGZlO2IXjLJ69C+7LmWW5eVeu3RRwlOIUzuAAPrqEO99CAFjBAeIZXeHMenRfn3flYtK45xcwJ/JHz+QPnv4z9</latexit>✓

<latexit sha1_base64="+2a+YdAkK5gXAKPWKNIZ/6djIOk=">AAAB7XicbZBNSwMxEIZn61etX1WPXoJF8FR2RdBj0YvHCvYD2qVk07SNzSZLMiuUpf/BiwdFvPp/vPlvTNs9aOsLgYd3ZsjMGyVSWPT9b6+wtr6xuVXcLu3s7u0flA+PmlanhvEG01KbdkQtl0LxBgqUvJ0YTuNI8lY0vp3VW0/cWKHVA04SHsZ0qMRAMIrOanZxxJH2yhW/6s9FViHIoQK56r3yV7evWRpzhUxSazuBn2CYUYOCST4tdVPLE8rGdMg7DhWNuQ2z+bZTcuacPhlo455CMnd/T2Q0tnYSR64zpjiyy7WZ+V+tk+LgOsyESlLkii0+GqSSoCaz00lfGM5QThxQZoTblbARNZShC6jkQgiWT16F5kU1cHx/Wand5HEU4QRO4RwCuIIa3EEdGsDgEZ7hFd487b14797HorXg5TPH8Efe5w+j+Y8o</latexit><latexit sha1_base64="+2a+YdAkK5gXAKPWKNIZ/6djIOk=">AAAB7XicbZBNSwMxEIZn61etX1WPXoJF8FR2RdBj0YvHCvYD2qVk07SNzSZLMiuUpf/BiwdFvPp/vPlvTNs9aOsLgYd3ZsjMGyVSWPT9b6+wtr6xuVXcLu3s7u0flA+PmlanhvEG01KbdkQtl0LxBgqUvJ0YTuNI8lY0vp3VW0/cWKHVA04SHsZ0qMRAMIrOanZxxJH2yhW/6s9FViHIoQK56r3yV7evWRpzhUxSazuBn2CYUYOCST4tdVPLE8rGdMg7DhWNuQ2z+bZTcuacPhlo455CMnd/T2Q0tnYSR64zpjiyy7WZ+V+tk+LgOsyESlLkii0+GqSSoCaz00lfGM5QThxQZoTblbARNZShC6jkQgiWT16F5kU1cHx/Wand5HEU4QRO4RwCuIIa3EEdGsDgEZ7hFd487b14797HorXg5TPH8Efe5w+j+Y8o</latexit><latexit sha1_base64="+2a+YdAkK5gXAKPWKNIZ/6djIOk=">AAAB7XicbZBNSwMxEIZn61etX1WPXoJF8FR2RdBj0YvHCvYD2qVk07SNzSZLMiuUpf/BiwdFvPp/vPlvTNs9aOsLgYd3ZsjMGyVSWPT9b6+wtr6xuVXcLu3s7u0flA+PmlanhvEG01KbdkQtl0LxBgqUvJ0YTuNI8lY0vp3VW0/cWKHVA04SHsZ0qMRAMIrOanZxxJH2yhW/6s9FViHIoQK56r3yV7evWRpzhUxSazuBn2CYUYOCST4tdVPLE8rGdMg7DhWNuQ2z+bZTcuacPhlo455CMnd/T2Q0tnYSR64zpjiyy7WZ+V+tk+LgOsyESlLkii0+GqSSoCaz00lfGM5QThxQZoTblbARNZShC6jkQgiWT16F5kU1cHx/Wand5HEU4QRO4RwCuIIa3EEdGsDgEZ7hFd487b14797HorXg5TPH8Efe5w+j+Y8o</latexit><latexit sha1_base64="+2a+YdAkK5gXAKPWKNIZ/6djIOk=">AAAB7XicbZBNSwMxEIZn61etX1WPXoJF8FR2RdBj0YvHCvYD2qVk07SNzSZLMiuUpf/BiwdFvPp/vPlvTNs9aOsLgYd3ZsjMGyVSWPT9b6+wtr6xuVXcLu3s7u0flA+PmlanhvEG01KbdkQtl0LxBgqUvJ0YTuNI8lY0vp3VW0/cWKHVA04SHsZ0qMRAMIrOanZxxJH2yhW/6s9FViHIoQK56r3yV7evWRpzhUxSazuBn2CYUYOCST4tdVPLE8rGdMg7DhWNuQ2z+bZTcuacPhlo455CMnd/T2Q0tnYSR64zpjiyy7WZ+V+tk+LgOsyESlLkii0+GqSSoCaz00lfGM5QThxQZoTblbARNZShC6jkQgiWT16F5kU1cHx/Wand5HEU4QRO4RwCuIIa3EEdGsDgEZ7hFd487b14797HorXg5TPH8Efe5w+j+Y8o</latexit>

kxk cos ✓<latexit sha1_base64="l948m21HmgBSkPWPzlTrden6Ts0=">AAAB+XicbZDNSgMxFIUz9a/Wv1GXboJFcFVmRNBl0Y3LCrYVOkPJpJk2NJMMyZ1imfZN3LhQxK1v4s63MW1noa0HAh/n3su9OVEquAHP+3ZKa+sbm1vl7crO7t7+gXt41DIq05Q1qRJKP0bEMMElawIHwR5TzUgSCdaOhrezenvEtOFKPsA4ZWFC+pLHnBKwVtd1g8lTMMEBVSaAAQPSdatezZsLr4JfQBUVanTdr6CnaJYwCVQQYzq+l0KYEw2cCjatBJlhKaFD0mcdi5IkzIT5/PIpPrNOD8dK2ycBz93fEzlJjBknke1MCAzMcm1m/lfrZBBfhzmXaQZM0sWiOBMYFJ7FgHtcMwpibIFQze2tmA6IJhRsWBUbgr/85VVoXdR8y/eX1fpNEUcZnaBTdI58dIXq6A41UBNRNELP6BW9Obnz4rw7H4vWklPMHKM/cj5/AKRGk6Y=</latexit><latexit sha1_base64="l948m21HmgBSkPWPzlTrden6Ts0=">AAAB+XicbZDNSgMxFIUz9a/Wv1GXboJFcFVmRNBl0Y3LCrYVOkPJpJk2NJMMyZ1imfZN3LhQxK1v4s63MW1noa0HAh/n3su9OVEquAHP+3ZKa+sbm1vl7crO7t7+gXt41DIq05Q1qRJKP0bEMMElawIHwR5TzUgSCdaOhrezenvEtOFKPsA4ZWFC+pLHnBKwVtd1g8lTMMEBVSaAAQPSdatezZsLr4JfQBUVanTdr6CnaJYwCVQQYzq+l0KYEw2cCjatBJlhKaFD0mcdi5IkzIT5/PIpPrNOD8dK2ycBz93fEzlJjBknke1MCAzMcm1m/lfrZBBfhzmXaQZM0sWiOBMYFJ7FgHtcMwpibIFQze2tmA6IJhRsWBUbgr/85VVoXdR8y/eX1fpNEUcZnaBTdI58dIXq6A41UBNRNELP6BW9Obnz4rw7H4vWklPMHKM/cj5/AKRGk6Y=</latexit><latexit sha1_base64="l948m21HmgBSkPWPzlTrden6Ts0=">AAAB+XicbZDNSgMxFIUz9a/Wv1GXboJFcFVmRNBl0Y3LCrYVOkPJpJk2NJMMyZ1imfZN3LhQxK1v4s63MW1noa0HAh/n3su9OVEquAHP+3ZKa+sbm1vl7crO7t7+gXt41DIq05Q1qRJKP0bEMMElawIHwR5TzUgSCdaOhrezenvEtOFKPsA4ZWFC+pLHnBKwVtd1g8lTMMEBVSaAAQPSdatezZsLr4JfQBUVanTdr6CnaJYwCVQQYzq+l0KYEw2cCjatBJlhKaFD0mcdi5IkzIT5/PIpPrNOD8dK2ycBz93fEzlJjBknke1MCAzMcm1m/lfrZBBfhzmXaQZM0sWiOBMYFJ7FgHtcMwpibIFQze2tmA6IJhRsWBUbgr/85VVoXdR8y/eX1fpNEUcZnaBTdI58dIXq6A41UBNRNELP6BW9Obnz4rw7H4vWklPMHKM/cj5/AKRGk6Y=</latexit><latexit sha1_base64="l948m21HmgBSkPWPzlTrden6Ts0=">AAAB+XicbZDNSgMxFIUz9a/Wv1GXboJFcFVmRNBl0Y3LCrYVOkPJpJk2NJMMyZ1imfZN3LhQxK1v4s63MW1noa0HAh/n3su9OVEquAHP+3ZKa+sbm1vl7crO7t7+gXt41DIq05Q1qRJKP0bEMMElawIHwR5TzUgSCdaOhrezenvEtOFKPsA4ZWFC+pLHnBKwVtd1g8lTMMEBVSaAAQPSdatezZsLr4JfQBUVanTdr6CnaJYwCVQQYzq+l0KYEw2cCjatBJlhKaFD0mcdi5IkzIT5/PIpPrNOD8dK2ycBz93fEzlJjBknke1MCAzMcm1m/lfrZBBfhzmXaQZM0sWiOBMYFJ7FgHtcMwpibIFQze2tmA6IJhRsWBUbgr/85VVoXdR8y/eX1fpNEUcZnaBTdI58dIXq6A41UBNRNELP6BW9Obnz4rw7H4vWklPMHKM/cj5/AKRGk6Y=</latexit>

hx, yi = kxk · kyk · cos ✓<latexit sha1_base64="Erkef27IuCLz13CGlmdw2LKnN48=">AAACL3icbZDLSgMxFIYz3q23qks3wSK4EJkRQTeCKIhLBatCU0omPdMGM8mQnJEOrW/kxldxI6KIW9/C9AJeDwQ+vnNCcv44U9JhGD4HY+MTk1PTM7OlufmFxaXy8sqlM7kVUBVGGXsdcwdKaqiiRAXXmQWexgqu4pvjfv/qFqyTRl9gkUE95S0tEyk4etUonzAFCVKmuG4poJ0tWlBmZavtnR26A8p6HdajTDSNt73ii4VxDNuAvFGuhNvhoOhfiEZQIaM6a5QfWdOIPAWNQnHnalGYYb3LLUqh4K7EcgcZFze8BTWPmqfg6t3Bvnd0w5smTYz1RyMd2O83ujx1rkhjP5lybLvfvb78r1fLMdmvd6XOcgQthg8luaJoaD882pQWBKrCAxdW+r9S0eaWC/QRl3wI0e+V/8Llznbk+Xy3cng0imOGrJF1skkiskcOySk5I1UiyD15JC/kNXgInoK34H04OhaM7qySHxV8fAJp7Ki2</latexit><latexit sha1_base64="Erkef27IuCLz13CGlmdw2LKnN48=">AAACL3icbZDLSgMxFIYz3q23qks3wSK4EJkRQTeCKIhLBatCU0omPdMGM8mQnJEOrW/kxldxI6KIW9/C9AJeDwQ+vnNCcv44U9JhGD4HY+MTk1PTM7OlufmFxaXy8sqlM7kVUBVGGXsdcwdKaqiiRAXXmQWexgqu4pvjfv/qFqyTRl9gkUE95S0tEyk4etUonzAFCVKmuG4poJ0tWlBmZavtnR26A8p6HdajTDSNt73ii4VxDNuAvFGuhNvhoOhfiEZQIaM6a5QfWdOIPAWNQnHnalGYYb3LLUqh4K7EcgcZFze8BTWPmqfg6t3Bvnd0w5smTYz1RyMd2O83ujx1rkhjP5lybLvfvb78r1fLMdmvd6XOcgQthg8luaJoaD882pQWBKrCAxdW+r9S0eaWC/QRl3wI0e+V/8Llznbk+Xy3cng0imOGrJF1skkiskcOySk5I1UiyD15JC/kNXgInoK34H04OhaM7qySHxV8fAJp7Ki2</latexit><latexit sha1_base64="Erkef27IuCLz13CGlmdw2LKnN48=">AAACL3icbZDLSgMxFIYz3q23qks3wSK4EJkRQTeCKIhLBatCU0omPdMGM8mQnJEOrW/kxldxI6KIW9/C9AJeDwQ+vnNCcv44U9JhGD4HY+MTk1PTM7OlufmFxaXy8sqlM7kVUBVGGXsdcwdKaqiiRAXXmQWexgqu4pvjfv/qFqyTRl9gkUE95S0tEyk4etUonzAFCVKmuG4poJ0tWlBmZavtnR26A8p6HdajTDSNt73ii4VxDNuAvFGuhNvhoOhfiEZQIaM6a5QfWdOIPAWNQnHnalGYYb3LLUqh4K7EcgcZFze8BTWPmqfg6t3Bvnd0w5smTYz1RyMd2O83ujx1rkhjP5lybLvfvb78r1fLMdmvd6XOcgQthg8luaJoaD882pQWBKrCAxdW+r9S0eaWC/QRl3wI0e+V/8Llznbk+Xy3cng0imOGrJF1skkiskcOySk5I1UiyD15JC/kNXgInoK34H04OhaM7qySHxV8fAJp7Ki2</latexit><latexit sha1_base64="Erkef27IuCLz13CGlmdw2LKnN48=">AAACL3icbZDLSgMxFIYz3q23qks3wSK4EJkRQTeCKIhLBatCU0omPdMGM8mQnJEOrW/kxldxI6KIW9/C9AJeDwQ+vnNCcv44U9JhGD4HY+MTk1PTM7OlufmFxaXy8sqlM7kVUBVGGXsdcwdKaqiiRAXXmQWexgqu4pvjfv/qFqyTRl9gkUE95S0tEyk4etUonzAFCVKmuG4poJ0tWlBmZavtnR26A8p6HdajTDSNt73ii4VxDNuAvFGuhNvhoOhfiEZQIaM6a5QfWdOIPAWNQnHnalGYYb3LLUqh4K7EcgcZFze8BTWPmqfg6t3Bvnd0w5smTYz1RyMd2O83ujx1rkhjP5lybLvfvb78r1fLMdmvd6XOcgQthg8luaJoaD882pQWBKrCAxdW+r9S0eaWC/QRl3wI0e+V/8Llznbk+Xy3cng0imOGrJF1skkiskcOySk5I1UiyD15JC/kNXgInoK34H04OhaM7qySHxV8fAJp7Ki2</latexit>

Page 115: COMP 414/514: Optimization - Algorithms, Complexity and

Vectors- Norms = notion of distance in multiple dimensions

kxk � 0, 8x 2 Rp

kxk = 0, i↵ x = 0

k↵xk = |↵|kxk, 8↵ 2 Rkx+ yk kxk+ kyk|x>y| kxkkyk

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Properties: (Triangle inequality)

(Cauchy-Schwarz)

Page 116: COMP 414/514: Optimization - Algorithms, Complexity and

Vectors- Norms = notion of distance in multiple dimensions

kxk � 0, 8x 2 Rp

kxk = 0, i↵ x = 0

k↵xk = |↵|kxk, 8↵ 2 Rkx+ yk kxk+ kyk|x>y| kxkkyk

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- Standard vector norms:

kxk2 =

sX

i

x2i

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kxk1 =X

i

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kxk1 = maxi

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Properties: (Triangle inequality)

(Cauchy-Schwarz)

Page 117: COMP 414/514: Optimization - Algorithms, Complexity and

Vectors- Norms = notion of distance in multiple dimensions

kxk � 0, 8x 2 Rp

kxk = 0, i↵ x = 0

k↵xk = |↵|kxk, 8↵ 2 Rkx+ yk kxk+ kyk|x>y| kxkkyk

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- Standard vector norms:

kxk2 =

sX

i

x2i

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kxk1 =X

i

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kxk1 = maxi

|xi|<latexit sha1_base64="ZwiIv/rroBMMfIMttUrsCBYwCTw=">AAACBnicbVDLSgMxFM34rPU16lKEYBFclRkRdCMU3bisYB/QGYZMmmlDk8yQZKTDtCs3/oobF4q49Rvc+Tem7Sy09cC9HM65l+SeMGFUacf5tpaWV1bX1ksb5c2t7Z1de2+/qeJUYtLAMYtlO0SKMCpIQ1PNSDuRBPGQkVY4uJn4rQciFY3Fvc4S4nPUEzSiGGkjBfaRNxp6oyD3qIh0NoZX0ONoGFA4Mm0U2BWn6kwBF4lbkAooUA/sL68b45QToTFDSnVcJ9F+jqSmmJFx2UsVSRAeoB7pGCoQJ8rPp2eM4YlRujCKpSmh4VT9vZEjrlTGQzPJke6reW8i/ud1Uh1d+jkVSaqJwLOHopRBHcNJJrBLJcGaZYYgLKn5K8R9JBHWJrmyCcGdP3mRNM+qrlN1784rtesijhI4BMfgFLjgAtTALaiDBsDgETyDV/BmPVkv1rv1MRtdsoqdA/AH1ucPapyZEQ==</latexit><latexit sha1_base64="ZwiIv/rroBMMfIMttUrsCBYwCTw=">AAACBnicbVDLSgMxFM34rPU16lKEYBFclRkRdCMU3bisYB/QGYZMmmlDk8yQZKTDtCs3/oobF4q49Rvc+Tem7Sy09cC9HM65l+SeMGFUacf5tpaWV1bX1ksb5c2t7Z1de2+/qeJUYtLAMYtlO0SKMCpIQ1PNSDuRBPGQkVY4uJn4rQciFY3Fvc4S4nPUEzSiGGkjBfaRNxp6oyD3qIh0NoZX0ONoGFA4Mm0U2BWn6kwBF4lbkAooUA/sL68b45QToTFDSnVcJ9F+jqSmmJFx2UsVSRAeoB7pGCoQJ8rPp2eM4YlRujCKpSmh4VT9vZEjrlTGQzPJke6reW8i/ud1Uh1d+jkVSaqJwLOHopRBHcNJJrBLJcGaZYYgLKn5K8R9JBHWJrmyCcGdP3mRNM+qrlN1784rtesijhI4BMfgFLjgAtTALaiDBsDgETyDV/BmPVkv1rv1MRtdsoqdA/AH1ucPapyZEQ==</latexit><latexit sha1_base64="ZwiIv/rroBMMfIMttUrsCBYwCTw=">AAACBnicbVDLSgMxFM34rPU16lKEYBFclRkRdCMU3bisYB/QGYZMmmlDk8yQZKTDtCs3/oobF4q49Rvc+Tem7Sy09cC9HM65l+SeMGFUacf5tpaWV1bX1ksb5c2t7Z1de2+/qeJUYtLAMYtlO0SKMCpIQ1PNSDuRBPGQkVY4uJn4rQciFY3Fvc4S4nPUEzSiGGkjBfaRNxp6oyD3qIh0NoZX0ONoGFA4Mm0U2BWn6kwBF4lbkAooUA/sL68b45QToTFDSnVcJ9F+jqSmmJFx2UsVSRAeoB7pGCoQJ8rPp2eM4YlRujCKpSmh4VT9vZEjrlTGQzPJke6reW8i/ud1Uh1d+jkVSaqJwLOHopRBHcNJJrBLJcGaZYYgLKn5K8R9JBHWJrmyCcGdP3mRNM+qrlN1784rtesijhI4BMfgFLjgAtTALaiDBsDgETyDV/BmPVkv1rv1MRtdsoqdA/AH1ucPapyZEQ==</latexit><latexit sha1_base64="ZwiIv/rroBMMfIMttUrsCBYwCTw=">AAACBnicbVDLSgMxFM34rPU16lKEYBFclRkRdCMU3bisYB/QGYZMmmlDk8yQZKTDtCs3/oobF4q49Rvc+Tem7Sy09cC9HM65l+SeMGFUacf5tpaWV1bX1ksb5c2t7Z1de2+/qeJUYtLAMYtlO0SKMCpIQ1PNSDuRBPGQkVY4uJn4rQciFY3Fvc4S4nPUEzSiGGkjBfaRNxp6oyD3qIh0NoZX0ONoGFA4Mm0U2BWn6kwBF4lbkAooUA/sL68b45QToTFDSnVcJ9F+jqSmmJFx2UsVSRAeoB7pGCoQJ8rPp2eM4YlRujCKpSmh4VT9vZEjrlTGQzPJke6reW8i/ud1Uh1d+jkVSaqJwLOHopRBHcNJJrBLJcGaZYYgLKn5K8R9JBHWJrmyCcGdP3mRNM+qrlN1784rtesijhI4BMfgFLjgAtTALaiDBsDgETyDV/BmPVkv1rv1MRtdsoqdA/AH1ucPapyZEQ==</latexit>

Properties:

- Famous wanna-be norms: kxk0 = card(x)<latexit sha1_base64="V8N2LDBf8pfyaovMMrDvIIdFRV8=">AAACAXicbVDLSsNAFJ34rPUVdSO4GSxC3ZREBN0IRTcuK9gHNCFMJtN26OTBzI20pHXjr7hxoYhb/8Kdf+O0zUJbD1w4nHMv997jJ4IrsKxvY2l5ZXVtvbBR3Nza3tk19/YbKk4lZXUai1i2fKKY4BGrAwfBWolkJPQFa/r9m4nffGBS8Ti6h2HC3JB0I97hlICWPPPQGQ2ckWfhK+wAG0BGiQzG5cGpZ5asijUFXiR2TkooR80zv5wgpmnIIqCCKNW2rQTcjEjgVLBx0UkVSwjtky5raxqRkCk3m34wxidaCXAnlroiwFP190RGQqWGoa87QwI9Ne9NxP+8dgqdSzfjUZICi+hsUScVGGI8iQMHXDIKYqgJoZLrWzHtEUko6NCKOgR7/uVF0jir2FbFvjsvVa/zOAroCB2jMrLRBaqiW1RDdUTRI3pGr+jNeDJejHfjY9a6ZOQzB+gPjM8fi8aWSw==</latexit><latexit sha1_base64="V8N2LDBf8pfyaovMMrDvIIdFRV8=">AAACAXicbVDLSsNAFJ34rPUVdSO4GSxC3ZREBN0IRTcuK9gHNCFMJtN26OTBzI20pHXjr7hxoYhb/8Kdf+O0zUJbD1w4nHMv997jJ4IrsKxvY2l5ZXVtvbBR3Nza3tk19/YbKk4lZXUai1i2fKKY4BGrAwfBWolkJPQFa/r9m4nffGBS8Ti6h2HC3JB0I97hlICWPPPQGQ2ckWfhK+wAG0BGiQzG5cGpZ5asijUFXiR2TkooR80zv5wgpmnIIqCCKNW2rQTcjEjgVLBx0UkVSwjtky5raxqRkCk3m34wxidaCXAnlroiwFP190RGQqWGoa87QwI9Ne9NxP+8dgqdSzfjUZICi+hsUScVGGI8iQMHXDIKYqgJoZLrWzHtEUko6NCKOgR7/uVF0jir2FbFvjsvVa/zOAroCB2jMrLRBaqiW1RDdUTRI3pGr+jNeDJejHfjY9a6ZOQzB+gPjM8fi8aWSw==</latexit><latexit sha1_base64="V8N2LDBf8pfyaovMMrDvIIdFRV8=">AAACAXicbVDLSsNAFJ34rPUVdSO4GSxC3ZREBN0IRTcuK9gHNCFMJtN26OTBzI20pHXjr7hxoYhb/8Kdf+O0zUJbD1w4nHMv997jJ4IrsKxvY2l5ZXVtvbBR3Nza3tk19/YbKk4lZXUai1i2fKKY4BGrAwfBWolkJPQFa/r9m4nffGBS8Ti6h2HC3JB0I97hlICWPPPQGQ2ckWfhK+wAG0BGiQzG5cGpZ5asijUFXiR2TkooR80zv5wgpmnIIqCCKNW2rQTcjEjgVLBx0UkVSwjtky5raxqRkCk3m34wxidaCXAnlroiwFP190RGQqWGoa87QwI9Ne9NxP+8dgqdSzfjUZICi+hsUScVGGI8iQMHXDIKYqgJoZLrWzHtEUko6NCKOgR7/uVF0jir2FbFvjsvVa/zOAroCB2jMrLRBaqiW1RDdUTRI3pGr+jNeDJejHfjY9a6ZOQzB+gPjM8fi8aWSw==</latexit><latexit sha1_base64="V8N2LDBf8pfyaovMMrDvIIdFRV8=">AAACAXicbVDLSsNAFJ34rPUVdSO4GSxC3ZREBN0IRTcuK9gHNCFMJtN26OTBzI20pHXjr7hxoYhb/8Kdf+O0zUJbD1w4nHMv997jJ4IrsKxvY2l5ZXVtvbBR3Nza3tk19/YbKk4lZXUai1i2fKKY4BGrAwfBWolkJPQFa/r9m4nffGBS8Ti6h2HC3JB0I97hlICWPPPQGQ2ckWfhK+wAG0BGiQzG5cGpZ5asijUFXiR2TkooR80zv5wgpmnIIqCCKNW2rQTcjEjgVLBx0UkVSwjtky5raxqRkCk3m34wxidaCXAnlroiwFP190RGQqWGoa87QwI9Ne9NxP+8dgqdSzfjUZICi+hsUScVGGI8iQMHXDIKYqgJoZLrWzHtEUko6NCKOgR7/uVF0jir2FbFvjsvVa/zOAroCB2jMrLRBaqiW1RDdUTRI3pGr+jNeDJejHfjY9a6ZOQzB+gPjM8fi8aWSw==</latexit>

(Triangle inequality)

(Cauchy-Schwarz)

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Matrices- Matrix in m, n dimensions: A 2 Rm⇥n

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A =

2

6664

A11 A12 · · · A1n

A21 A22 · · · A2n...

. . ....

Am1 Am2 · · · Amn

3

7775

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Page 119: COMP 414/514: Optimization - Algorithms, Complexity and

Matrices- Matrix in m, n dimensions: A 2 Rm⇥n

<latexit sha1_base64="MxwCSL/gyMr0pbYAXqcODXgj2Pw=">AAACBnicbVBNS8NAEJ34WetX1KMIi0XwVBIR9Fj14rGK/YAmls120y7dbMLuRighJy/+FS8eFPHqb/Dmv3HT9qCtDwYe780wMy9IOFPacb6thcWl5ZXV0lp5fWNza9ve2W2qOJWENkjMY9kOsKKcCdrQTHPaTiTFUcBpKxheFX7rgUrFYnGnRwn1I9wXLGQEayN17YML5DGBvAjrQRBkt/l9FiFPs4gqJPKuXXGqzhhonrhTUoEp6l37y+vFJI2o0IRjpTquk2g/w1Izwmle9lJFE0yGuE87hgps9vjZ+I0cHRmlh8JYmhIajdXfExmOlBpFgekszlWzXiH+53VSHZ77GRNJqqkgk0VhypGOUZEJ6jFJieYjQzCRzNyKyABLTLRJrmxCcGdfnifNk6pr+M1ppXY5jaME+3AIx+DCGdTgGurQAAKP8Ayv8GY9WS/Wu/UxaV2wpjN78AfW5w+D2JiE</latexit><latexit sha1_base64="MxwCSL/gyMr0pbYAXqcODXgj2Pw=">AAACBnicbVBNS8NAEJ34WetX1KMIi0XwVBIR9Fj14rGK/YAmls120y7dbMLuRighJy/+FS8eFPHqb/Dmv3HT9qCtDwYe780wMy9IOFPacb6thcWl5ZXV0lp5fWNza9ve2W2qOJWENkjMY9kOsKKcCdrQTHPaTiTFUcBpKxheFX7rgUrFYnGnRwn1I9wXLGQEayN17YML5DGBvAjrQRBkt/l9FiFPs4gqJPKuXXGqzhhonrhTUoEp6l37y+vFJI2o0IRjpTquk2g/w1Izwmle9lJFE0yGuE87hgps9vjZ+I0cHRmlh8JYmhIajdXfExmOlBpFgekszlWzXiH+53VSHZ77GRNJqqkgk0VhypGOUZEJ6jFJieYjQzCRzNyKyABLTLRJrmxCcGdfnifNk6pr+M1ppXY5jaME+3AIx+DCGdTgGurQAAKP8Ayv8GY9WS/Wu/UxaV2wpjN78AfW5w+D2JiE</latexit><latexit sha1_base64="MxwCSL/gyMr0pbYAXqcODXgj2Pw=">AAACBnicbVBNS8NAEJ34WetX1KMIi0XwVBIR9Fj14rGK/YAmls120y7dbMLuRighJy/+FS8eFPHqb/Dmv3HT9qCtDwYe780wMy9IOFPacb6thcWl5ZXV0lp5fWNza9ve2W2qOJWENkjMY9kOsKKcCdrQTHPaTiTFUcBpKxheFX7rgUrFYnGnRwn1I9wXLGQEayN17YML5DGBvAjrQRBkt/l9FiFPs4gqJPKuXXGqzhhonrhTUoEp6l37y+vFJI2o0IRjpTquk2g/w1Izwmle9lJFE0yGuE87hgps9vjZ+I0cHRmlh8JYmhIajdXfExmOlBpFgekszlWzXiH+53VSHZ77GRNJqqkgk0VhypGOUZEJ6jFJieYjQzCRzNyKyABLTLRJrmxCcGdfnifNk6pr+M1ppXY5jaME+3AIx+DCGdTgGurQAAKP8Ayv8GY9WS/Wu/UxaV2wpjN78AfW5w+D2JiE</latexit><latexit sha1_base64="MxwCSL/gyMr0pbYAXqcODXgj2Pw=">AAACBnicbVBNS8NAEJ34WetX1KMIi0XwVBIR9Fj14rGK/YAmls120y7dbMLuRighJy/+FS8eFPHqb/Dmv3HT9qCtDwYe780wMy9IOFPacb6thcWl5ZXV0lp5fWNza9ve2W2qOJWENkjMY9kOsKKcCdrQTHPaTiTFUcBpKxheFX7rgUrFYnGnRwn1I9wXLGQEayN17YML5DGBvAjrQRBkt/l9FiFPs4gqJPKuXXGqzhhonrhTUoEp6l37y+vFJI2o0IRjpTquk2g/w1Izwmle9lJFE0yGuE87hgps9vjZ+I0cHRmlh8JYmhIajdXfExmOlBpFgekszlWzXiH+53VSHZ77GRNJqqkgk0VhypGOUZEJ6jFJieYjQzCRzNyKyABLTLRJrmxCcGdfnifNk6pr+M1ppXY5jaME+3AIx+DCGdTgGurQAAKP8Ayv8GY9WS/Wu/UxaV2wpjN78AfW5w+D2JiE</latexit>

A =

2

6664

A11 A12 · · · A1n

A21 A22 · · · A2n...

. . ....

Am1 Am2 · · · Amn

3

7775

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- Names: Square, tall, fat, zero, identity, diagonal

Page 120: COMP 414/514: Optimization - Algorithms, Complexity and

Matrices- Matrix in m, n dimensions: A 2 Rm⇥n

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A =

2

6664

A11 A12 · · · A1n

A21 A22 · · · A2n...

. . ....

Am1 Am2 · · · Amn

3

7775

<latexit sha1_base64="ieGb1JmyBV2I55sToSwTGJNADeY=">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</latexit><latexit sha1_base64="ieGb1JmyBV2I55sToSwTGJNADeY=">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</latexit><latexit sha1_base64="ieGb1JmyBV2I55sToSwTGJNADeY=">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</latexit><latexit sha1_base64="ieGb1JmyBV2I55sToSwTGJNADeY=">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</latexit>

- Names: Square, tall, fat, zero, identity, diagonal

- Properties:

A+B = B +A, 8A,B 2 Rm⇥n

(A+B) + C = A+ (B + C), 8A,B,C 2 Rm⇥n

A+ 0 = 0 +A, 8A 2 Rm⇥n

(A+B)> = A> +B>, 8A,B 2 Rm⇥n<latexit sha1_base64="gPB9DfbR9R5TLw3fFm/nRbobVPw=">AAAC93iclVJNbxMxEPUulJbQ0qQcuYyIqFK1qnYRUnup1CYXjgWRtFKcRl7Haa14vSt7tlK0Cj+ECweqiit/hRv/BnuTAyRIlJFsPb+ZeX7+SHIlLUbRzyB89HjtyfrG09qzza3n2/XGTs9mheGiyzOVmcuEWaGkFl2UqMRlbgRLEyUukknH5y9uhbEy0x9xmotByq61HEvO0FHDRrB1BvvQht0TN+3D2QF8ouPMMKU8bgOVGmjK8CZJyg+zqzIFijIVFvQMKK21qu49N3W8hF+1vE5nb0nowBX8Q8s3R14lWjbyQBdXFLO8sjFHjqvAf5xpWG9Gh1EVsAriBWiSRZwP6z/oKONFKjRyxaztx1GOg5IZlFyJWY0WVuSMT9i16DuomdtnUFbvNoPXjhmBs+aGRqjY3ztKllo7TRNX6e3a5Zwn/5brFzg+HpRS5wUKzecbjQsFmIH/BDCSRnBUUwcYN9J5BX7DDOPovkrNXUK8fORV0HtzGDv8/m3ztL24jg3ykrwiLRKTI3JK3pFz0iU8wOBz8DW4C6fhl/A+/DYvDYNFzwvyR4TffwE1FN5i</latexit><latexit 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Page 121: COMP 414/514: Optimization - Algorithms, Complexity and

Matrices- Matrix multiplication: where , , and C = AB

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B 2 Rn⇥p<latexit sha1_base64="EXB3w9wWi16rdVHdgvQBZlzMJFo=">AAACBnicbVDLSsNAFL3xWesr6lKEwSK4KokIuix147KKfUATy2Q6aYdOJmFmIpSQlRt/xY0LRdz6De78GydtF9p6YODMOfdy7z1BwpnSjvNtLS2vrK6tlzbKm1vbO7v23n5LxakktEliHstOgBXlTNCmZprTTiIpjgJO28HoqvDbD1QqFos7PU6oH+GBYCEjWBupZx/VkccE8iKsh0GQ3eb3mflpFlGFkrxnV5yqMwFaJO6MVGCGRs/+8voxSSMqNOFYqa7rJNrPsNSMcJqXvVTRBJMRHtCuoQKbOX42OSNHJ0bpozCW5gmNJurvjgxHSo2jwFQW66p5rxD/87qpDi/9jIkk1VSQ6aAw5UjHqMgE9ZmkRPOxIZhIZnZFZIglJtokVzYhuPMnL5LWWdU1/Oa8UqvP4ijBIRzDKbhwATW4hgY0gcAjPMMrvFlP1ov1bn1MS5esWc8B/IH1+QOKEJiI</latexit><latexit sha1_base64="EXB3w9wWi16rdVHdgvQBZlzMJFo=">AAACBnicbVDLSsNAFL3xWesr6lKEwSK4KokIuix147KKfUATy2Q6aYdOJmFmIpSQlRt/xY0LRdz6De78GydtF9p6YODMOfdy7z1BwpnSjvNtLS2vrK6tlzbKm1vbO7v23n5LxakktEliHstOgBXlTNCmZprTTiIpjgJO28HoqvDbD1QqFos7PU6oH+GBYCEjWBupZx/VkccE8iKsh0GQ3eb3mflpFlGFkrxnV5yqMwFaJO6MVGCGRs/+8voxSSMqNOFYqa7rJNrPsNSMcJqXvVTRBJMRHtCuoQKbOX42OSNHJ0bpozCW5gmNJurvjgxHSo2jwFQW66p5rxD/87qpDi/9jIkk1VSQ6aAw5UjHqMgE9ZmkRPOxIZhIZnZFZIglJtokVzYhuPMnL5LWWdU1/Oa8UqvP4ijBIRzDKbhwATW4hgY0gcAjPMMrvFlP1ov1bn1MS5esWc8B/IH1+QOKEJiI</latexit><latexit sha1_base64="EXB3w9wWi16rdVHdgvQBZlzMJFo=">AAACBnicbVDLSsNAFL3xWesr6lKEwSK4KokIuix147KKfUATy2Q6aYdOJmFmIpSQlRt/xY0LRdz6De78GydtF9p6YODMOfdy7z1BwpnSjvNtLS2vrK6tlzbKm1vbO7v23n5LxakktEliHstOgBXlTNCmZprTTiIpjgJO28HoqvDbD1QqFos7PU6oH+GBYCEjWBupZx/VkccE8iKsh0GQ3eb3mflpFlGFkrxnV5yqMwFaJO6MVGCGRs/+8voxSSMqNOFYqa7rJNrPsNSMcJqXvVTRBJMRHtCuoQKbOX42OSNHJ0bpozCW5gmNJurvjgxHSo2jwFQW66p5rxD/87qpDi/9jIkk1VSQ6aAw5UjHqMgE9ZmkRPOxIZhIZnZFZIglJtokVzYhuPMnL5LWWdU1/Oa8UqvP4ijBIRzDKbhwATW4hgY0gcAjPMMrvFlP1ov1bn1MS5esWc8B/IH1+QOKEJiI</latexit><latexit sha1_base64="EXB3w9wWi16rdVHdgvQBZlzMJFo=">AAACBnicbVDLSsNAFL3xWesr6lKEwSK4KokIuix147KKfUATy2Q6aYdOJmFmIpSQlRt/xY0LRdz6De78GydtF9p6YODMOfdy7z1BwpnSjvNtLS2vrK6tlzbKm1vbO7v23n5LxakktEliHstOgBXlTNCmZprTTiIpjgJO28HoqvDbD1QqFos7PU6oH+GBYCEjWBupZx/VkccE8iKsh0GQ3eb3mflpFlGFkrxnV5yqMwFaJO6MVGCGRs/+8voxSSMqNOFYqa7rJNrPsNSMcJqXvVTRBJMRHtCuoQKbOX42OSNHJ0bpozCW5gmNJurvjgxHSo2jwFQW66p5rxD/87qpDi/9jIkk1VSQ6aAw5UjHqMgE9ZmkRPOxIZhIZnZFZIglJtokVzYhuPMnL5LWWdU1/Oa8UqvP4ijBIRzDKbhwATW4hgY0gcAjPMMrvFlP1ov1bn1MS5esWc8B/IH1+QOKEJiI</latexit>

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6664

C11 C12 · · · C1p

C21 C22 · · · C2p...

. . ....

Cm1 Cm2 · · · Cmp

3

7775= C = AB =

2

6664

A11 A12 · · · A1n

A21 A22 · · · A2n...

. . ....

Am1 Am2 · · · Amn

3

7775·

2

6664

B11 B12 · · · B1p

B21 B22 · · · B2p...

. . ....

Bn1 Bn2 · · · Bnp

3

7775

Page 122: COMP 414/514: Optimization - Algorithms, Complexity and

Matrices- Matrix multiplication: where , , and

- Special cases: vector inner product, matrix-vector mult., outer product

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C 2 Rm⇥p<latexit sha1_base64="+CqQuVHMZT19nMiKQttiPx5YtbA=">AAACBnicbVBNS8NAEJ34WetX1KMIi0XwVBIR9FjsxWMV+wFNLJvtpl262YTdjVBCTl78K148KOLV3+DNf+Om7UFbHww83pthZl6QcKa043xbS8srq2vrpY3y5tb2zq69t99ScSoJbZKYx7ITYEU5E7Spmea0k0iKo4DTdjCqF377gUrFYnGnxwn1IzwQLGQEayP17KM68phAXoT1MAiy2/w+i5CnWUQVSvKeXXGqzgRokbgzUoEZGj37y+vHJI2o0IRjpbquk2g/w1Izwmle9lJFE0xGeEC7hgps9vjZ5I0cnRilj8JYmhIaTdTfExmOlBpHgekszlXzXiH+53VTHV76GRNJqqkg00VhypGOUZEJ6jNJieZjQzCRzNyKyBBLTLRJrmxCcOdfXiSts6pr+M15pXY1i6MEh3AMp+DCBdTgGhrQBAKP8Ayv8GY9WS/Wu/UxbV2yZjMH8AfW5w+KIpiI</latexit><latexit sha1_base64="+CqQuVHMZT19nMiKQttiPx5YtbA=">AAACBnicbVBNS8NAEJ34WetX1KMIi0XwVBIR9FjsxWMV+wFNLJvtpl262YTdjVBCTl78K148KOLV3+DNf+Om7UFbHww83pthZl6QcKa043xbS8srq2vrpY3y5tb2zq69t99ScSoJbZKYx7ITYEU5E7Spmea0k0iKo4DTdjCqF377gUrFYnGnxwn1IzwQLGQEayP17KM68phAXoT1MAiy2/w+i5CnWUQVSvKeXXGqzgRokbgzUoEZGj37y+vHJI2o0IRjpbquk2g/w1Izwmle9lJFE0xGeEC7hgps9vjZ5I0cnRilj8JYmhIaTdTfExmOlBpHgekszlXzXiH+53VTHV76GRNJqqkg00VhypGOUZEJ6jNJieZjQzCRzNyKyBBLTLRJrmxCcOdfXiSts6pr+M15pXY1i6MEh3AMp+DCBdTgGhrQBAKP8Ayv8GY9WS/Wu/UxbV2yZjMH8AfW5w+KIpiI</latexit><latexit sha1_base64="+CqQuVHMZT19nMiKQttiPx5YtbA=">AAACBnicbVBNS8NAEJ34WetX1KMIi0XwVBIR9FjsxWMV+wFNLJvtpl262YTdjVBCTl78K148KOLV3+DNf+Om7UFbHww83pthZl6QcKa043xbS8srq2vrpY3y5tb2zq69t99ScSoJbZKYx7ITYEU5E7Spmea0k0iKo4DTdjCqF377gUrFYnGnxwn1IzwQLGQEayP17KM68phAXoT1MAiy2/w+i5CnWUQVSvKeXXGqzgRokbgzUoEZGj37y+vHJI2o0IRjpbquk2g/w1Izwmle9lJFE0xGeEC7hgps9vjZ5I0cnRilj8JYmhIaTdTfExmOlBpHgekszlXzXiH+53VTHV76GRNJqqkg00VhypGOUZEJ6jNJieZjQzCRzNyKyBBLTLRJrmxCcOdfXiSts6pr+M15pXY1i6MEh3AMp+DCBdTgGhrQBAKP8Ayv8GY9WS/Wu/UxbV2yZjMH8AfW5w+KIpiI</latexit><latexit sha1_base64="+CqQuVHMZT19nMiKQttiPx5YtbA=">AAACBnicbVBNS8NAEJ34WetX1KMIi0XwVBIR9FjsxWMV+wFNLJvtpl262YTdjVBCTl78K148KOLV3+DNf+Om7UFbHww83pthZl6QcKa043xbS8srq2vrpY3y5tb2zq69t99ScSoJbZKYx7ITYEU5E7Spmea0k0iKo4DTdjCqF377gUrFYnGnxwn1IzwQLGQEayP17KM68phAXoT1MAiy2/w+i5CnWUQVSvKeXXGqzgRokbgzUoEZGj37y+vHJI2o0IRjpbquk2g/w1Izwmle9lJFE0xGeEC7hgps9vjZ5I0cnRilj8JYmhIaTdTfExmOlBpHgekszlXzXiH+53VTHV76GRNJqqkg00VhypGOUZEJ6jNJieZjQzCRzNyKyBBLTLRJrmxCcOdfXiSts6pr+M15pXY1i6MEh3AMp+DCBdTgGhrQBAKP8Ayv8GY9WS/Wu/UxbV2yZjMH8AfW5w+KIpiI</latexit>

A 2 Rm⇥n<latexit sha1_base64="MxwCSL/gyMr0pbYAXqcODXgj2Pw=">AAACBnicbVBNS8NAEJ34WetX1KMIi0XwVBIR9Fj14rGK/YAmls120y7dbMLuRighJy/+FS8eFPHqb/Dmv3HT9qCtDwYe780wMy9IOFPacb6thcWl5ZXV0lp5fWNza9ve2W2qOJWENkjMY9kOsKKcCdrQTHPaTiTFUcBpKxheFX7rgUrFYnGnRwn1I9wXLGQEayN17YML5DGBvAjrQRBkt/l9FiFPs4gqJPKuXXGqzhhonrhTUoEp6l37y+vFJI2o0IRjpTquk2g/w1Izwmle9lJFE0yGuE87hgps9vjZ+I0cHRmlh8JYmhIajdXfExmOlBpFgekszlWzXiH+53VSHZ77GRNJqqkgk0VhypGOUZEJ6jFJieYjQzCRzNyKyABLTLRJrmxCcGdfnifNk6pr+M1ppXY5jaME+3AIx+DCGdTgGurQAAKP8Ayv8GY9WS/Wu/UxaV2wpjN78AfW5w+D2JiE</latexit><latexit sha1_base64="MxwCSL/gyMr0pbYAXqcODXgj2Pw=">AAACBnicbVBNS8NAEJ34WetX1KMIi0XwVBIR9Fj14rGK/YAmls120y7dbMLuRighJy/+FS8eFPHqb/Dmv3HT9qCtDwYe780wMy9IOFPacb6thcWl5ZXV0lp5fWNza9ve2W2qOJWENkjMY9kOsKKcCdrQTHPaTiTFUcBpKxheFX7rgUrFYnGnRwn1I9wXLGQEayN17YML5DGBvAjrQRBkt/l9FiFPs4gqJPKuXXGqzhhonrhTUoEp6l37y+vFJI2o0IRjpTquk2g/w1Izwmle9lJFE0yGuE87hgps9vjZ+I0cHRmlh8JYmhIajdXfExmOlBpFgekszlWzXiH+53VSHZ77GRNJqqkgk0VhypGOUZEJ6jFJieYjQzCRzNyKyABLTLRJrmxCcGdfnifNk6pr+M1ppXY5jaME+3AIx+DCGdTgGurQAAKP8Ayv8GY9WS/Wu/UxaV2wpjN78AfW5w+D2JiE</latexit><latexit sha1_base64="MxwCSL/gyMr0pbYAXqcODXgj2Pw=">AAACBnicbVBNS8NAEJ34WetX1KMIi0XwVBIR9Fj14rGK/YAmls120y7dbMLuRighJy/+FS8eFPHqb/Dmv3HT9qCtDwYe780wMy9IOFPacb6thcWl5ZXV0lp5fWNza9ve2W2qOJWENkjMY9kOsKKcCdrQTHPaTiTFUcBpKxheFX7rgUrFYnGnRwn1I9wXLGQEayN17YML5DGBvAjrQRBkt/l9FiFPs4gqJPKuXXGqzhhonrhTUoEp6l37y+vFJI2o0IRjpTquk2g/w1Izwmle9lJFE0yGuE87hgps9vjZ+I0cHRmlh8JYmhIajdXfExmOlBpFgekszlWzXiH+53VSHZ77GRNJqqkgk0VhypGOUZEJ6jFJieYjQzCRzNyKyABLTLRJrmxCcGdfnifNk6pr+M1ppXY5jaME+3AIx+DCGdTgGurQAAKP8Ayv8GY9WS/Wu/UxaV2wpjN78AfW5w+D2JiE</latexit><latexit sha1_base64="MxwCSL/gyMr0pbYAXqcODXgj2Pw=">AAACBnicbVBNS8NAEJ34WetX1KMIi0XwVBIR9Fj14rGK/YAmls120y7dbMLuRighJy/+FS8eFPHqb/Dmv3HT9qCtDwYe780wMy9IOFPacb6thcWl5ZXV0lp5fWNza9ve2W2qOJWENkjMY9kOsKKcCdrQTHPaTiTFUcBpKxheFX7rgUrFYnGnRwn1I9wXLGQEayN17YML5DGBvAjrQRBkt/l9FiFPs4gqJPKuXXGqzhhonrhTUoEp6l37y+vFJI2o0IRjpTquk2g/w1Izwmle9lJFE0yGuE87hgps9vjZ+I0cHRmlh8JYmhIajdXfExmOlBpFgekszlWzXiH+53VSHZ77GRNJqqkgk0VhypGOUZEJ6jFJieYjQzCRzNyKyABLTLRJrmxCcGdfnifNk6pr+M1ppXY5jaME+3AIx+DCGdTgGurQAAKP8Ayv8GY9WS/Wu/UxaV2wpjN78AfW5w+D2JiE</latexit>

B 2 Rn⇥p<latexit sha1_base64="EXB3w9wWi16rdVHdgvQBZlzMJFo=">AAACBnicbVDLSsNAFL3xWesr6lKEwSK4KokIuix147KKfUATy2Q6aYdOJmFmIpSQlRt/xY0LRdz6De78GydtF9p6YODMOfdy7z1BwpnSjvNtLS2vrK6tlzbKm1vbO7v23n5LxakktEliHstOgBXlTNCmZprTTiIpjgJO28HoqvDbD1QqFos7PU6oH+GBYCEjWBupZx/VkccE8iKsh0GQ3eb3mflpFlGFkrxnV5yqMwFaJO6MVGCGRs/+8voxSSMqNOFYqa7rJNrPsNSMcJqXvVTRBJMRHtCuoQKbOX42OSNHJ0bpozCW5gmNJurvjgxHSo2jwFQW66p5rxD/87qpDi/9jIkk1VSQ6aAw5UjHqMgE9ZmkRPOxIZhIZnZFZIglJtokVzYhuPMnL5LWWdU1/Oa8UqvP4ijBIRzDKbhwATW4hgY0gcAjPMMrvFlP1ov1bn1MS5esWc8B/IH1+QOKEJiI</latexit><latexit sha1_base64="EXB3w9wWi16rdVHdgvQBZlzMJFo=">AAACBnicbVDLSsNAFL3xWesr6lKEwSK4KokIuix147KKfUATy2Q6aYdOJmFmIpSQlRt/xY0LRdz6De78GydtF9p6YODMOfdy7z1BwpnSjvNtLS2vrK6tlzbKm1vbO7v23n5LxakktEliHstOgBXlTNCmZprTTiIpjgJO28HoqvDbD1QqFos7PU6oH+GBYCEjWBupZx/VkccE8iKsh0GQ3eb3mflpFlGFkrxnV5yqMwFaJO6MVGCGRs/+8voxSSMqNOFYqa7rJNrPsNSMcJqXvVTRBJMRHtCuoQKbOX42OSNHJ0bpozCW5gmNJurvjgxHSo2jwFQW66p5rxD/87qpDi/9jIkk1VSQ6aAw5UjHqMgE9ZmkRPOxIZhIZnZFZIglJtokVzYhuPMnL5LWWdU1/Oa8UqvP4ijBIRzDKbhwATW4hgY0gcAjPMMrvFlP1ov1bn1MS5esWc8B/IH1+QOKEJiI</latexit><latexit sha1_base64="EXB3w9wWi16rdVHdgvQBZlzMJFo=">AAACBnicbVDLSsNAFL3xWesr6lKEwSK4KokIuix147KKfUATy2Q6aYdOJmFmIpSQlRt/xY0LRdz6De78GydtF9p6YODMOfdy7z1BwpnSjvNtLS2vrK6tlzbKm1vbO7v23n5LxakktEliHstOgBXlTNCmZprTTiIpjgJO28HoqvDbD1QqFos7PU6oH+GBYCEjWBupZx/VkccE8iKsh0GQ3eb3mflpFlGFkrxnV5yqMwFaJO6MVGCGRs/+8voxSSMqNOFYqa7rJNrPsNSMcJqXvVTRBJMRHtCuoQKbOX42OSNHJ0bpozCW5gmNJurvjgxHSo2jwFQW66p5rxD/87qpDi/9jIkk1VSQ6aAw5UjHqMgE9ZmkRPOxIZhIZnZFZIglJtokVzYhuPMnL5LWWdU1/Oa8UqvP4ijBIRzDKbhwATW4hgY0gcAjPMMrvFlP1ov1bn1MS5esWc8B/IH1+QOKEJiI</latexit><latexit sha1_base64="EXB3w9wWi16rdVHdgvQBZlzMJFo=">AAACBnicbVDLSsNAFL3xWesr6lKEwSK4KokIuix147KKfUATy2Q6aYdOJmFmIpSQlRt/xY0LRdz6De78GydtF9p6YODMOfdy7z1BwpnSjvNtLS2vrK6tlzbKm1vbO7v23n5LxakktEliHstOgBXlTNCmZprTTiIpjgJO28HoqvDbD1QqFos7PU6oH+GBYCEjWBupZx/VkccE8iKsh0GQ3eb3mflpFlGFkrxnV5yqMwFaJO6MVGCGRs/+8voxSSMqNOFYqa7rJNrPsNSMcJqXvVTRBJMRHtCuoQKbOX42OSNHJ0bpozCW5gmNJurvjgxHSo2jwFQW66p5rxD/87qpDi/9jIkk1VSQ6aAw5UjHqMgE9ZmkRPOxIZhIZnZFZIglJtokVzYhuPMnL5LWWdU1/Oa8UqvP4ijBIRzDKbhwATW4hgY0gcAjPMMrvFlP1ov1bn1MS5esWc8B/IH1+QOKEJiI</latexit>

<latexit sha1_base64="hfRm/dkx6hM/0KpRtYEuroYj530=">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</latexit>2

6664

C11 C12 · · · C1p

C21 C22 · · · C2p...

. . ....

Cm1 Cm2 · · · Cmp

3

7775= C = AB =

2

6664

A11 A12 · · · A1n

A21 A22 · · · A2n...

. . ....

Am1 Am2 · · · Amn

3

7775·

2

6664

B11 B12 · · · B1p

B21 B22 · · · B2p...

. . ....

Bn1 Bn2 · · · Bnp

3

7775

Page 123: COMP 414/514: Optimization - Algorithms, Complexity and

Matrices- Matrix multiplication: where , , and

- Special cases: vector inner product, matrix-vector mult., outer product

C = AB<latexit sha1_base64="MpSDp7T1X4ao1rjnAJ3SssogNAI=">AAAB7XicbZDLSgMxFIbP1Futt6pLN8EiuCozIuhGqO3GZQV7gXYomTTTxmaSIckIZeg7uHGhiFvfx51vYzqdhbb+EPj4zznknD+IOdPGdb+dwtr6xuZWcbu0s7u3f1A+PGprmShCW0RyqboB1pQzQVuGGU67saI4CjjtBJPGvN55okozKR7MNKZ+hEeChYxgY612A92g2/qgXHGrbia0Cl4OFcjVHJS/+kNJkogKQzjWuue5sfFTrAwjnM5K/UTTGJMJHtGeRYEjqv0023aGzqwzRKFU9gmDMvf3RIojradRYDsjbMZ6uTY3/6v1EhNe+ykTcWKoIIuPwoQjI9H8dDRkihLDpxYwUczuisgYK0yMDahkQ/CWT16F9kXVs3x/WanV8ziKcAKncA4eXEEN7qAJLSDwCM/wCm+OdF6cd+dj0Vpw8plj+CPn8wfVlI35</latexit><latexit sha1_base64="MpSDp7T1X4ao1rjnAJ3SssogNAI=">AAAB7XicbZDLSgMxFIbP1Futt6pLN8EiuCozIuhGqO3GZQV7gXYomTTTxmaSIckIZeg7uHGhiFvfx51vYzqdhbb+EPj4zznknD+IOdPGdb+dwtr6xuZWcbu0s7u3f1A+PGprmShCW0RyqboB1pQzQVuGGU67saI4CjjtBJPGvN55okozKR7MNKZ+hEeChYxgY612A92g2/qgXHGrbia0Cl4OFcjVHJS/+kNJkogKQzjWuue5sfFTrAwjnM5K/UTTGJMJHtGeRYEjqv0023aGzqwzRKFU9gmDMvf3RIojradRYDsjbMZ6uTY3/6v1EhNe+ykTcWKoIIuPwoQjI9H8dDRkihLDpxYwUczuisgYK0yMDahkQ/CWT16F9kXVs3x/WanV8ziKcAKncA4eXEEN7qAJLSDwCM/wCm+OdF6cd+dj0Vpw8plj+CPn8wfVlI35</latexit><latexit sha1_base64="MpSDp7T1X4ao1rjnAJ3SssogNAI=">AAAB7XicbZDLSgMxFIbP1Futt6pLN8EiuCozIuhGqO3GZQV7gXYomTTTxmaSIckIZeg7uHGhiFvfx51vYzqdhbb+EPj4zznknD+IOdPGdb+dwtr6xuZWcbu0s7u3f1A+PGprmShCW0RyqboB1pQzQVuGGU67saI4CjjtBJPGvN55okozKR7MNKZ+hEeChYxgY612A92g2/qgXHGrbia0Cl4OFcjVHJS/+kNJkogKQzjWuue5sfFTrAwjnM5K/UTTGJMJHtGeRYEjqv0023aGzqwzRKFU9gmDMvf3RIojradRYDsjbMZ6uTY3/6v1EhNe+ykTcWKoIIuPwoQjI9H8dDRkihLDpxYwUczuisgYK0yMDahkQ/CWT16F9kXVs3x/WanV8ziKcAKncA4eXEEN7qAJLSDwCM/wCm+OdF6cd+dj0Vpw8plj+CPn8wfVlI35</latexit><latexit sha1_base64="MpSDp7T1X4ao1rjnAJ3SssogNAI=">AAAB7XicbZDLSgMxFIbP1Futt6pLN8EiuCozIuhGqO3GZQV7gXYomTTTxmaSIckIZeg7uHGhiFvfx51vYzqdhbb+EPj4zznknD+IOdPGdb+dwtr6xuZWcbu0s7u3f1A+PGprmShCW0RyqboB1pQzQVuGGU67saI4CjjtBJPGvN55okozKR7MNKZ+hEeChYxgY612A92g2/qgXHGrbia0Cl4OFcjVHJS/+kNJkogKQzjWuue5sfFTrAwjnM5K/UTTGJMJHtGeRYEjqv0023aGzqwzRKFU9gmDMvf3RIojradRYDsjbMZ6uTY3/6v1EhNe+ykTcWKoIIuPwoQjI9H8dDRkihLDpxYwUczuisgYK0yMDahkQ/CWT16F9kXVs3x/WanV8ziKcAKncA4eXEEN7qAJLSDwCM/wCm+OdF6cd+dj0Vpw8plj+CPn8wfVlI35</latexit>

C 2 Rm⇥p<latexit sha1_base64="+CqQuVHMZT19nMiKQttiPx5YtbA=">AAACBnicbVBNS8NAEJ34WetX1KMIi0XwVBIR9FjsxWMV+wFNLJvtpl262YTdjVBCTl78K148KOLV3+DNf+Om7UFbHww83pthZl6QcKa043xbS8srq2vrpY3y5tb2zq69t99ScSoJbZKYx7ITYEU5E7Spmea0k0iKo4DTdjCqF377gUrFYnGnxwn1IzwQLGQEayP17KM68phAXoT1MAiy2/w+i5CnWUQVSvKeXXGqzgRokbgzUoEZGj37y+vHJI2o0IRjpbquk2g/w1Izwmle9lJFE0xGeEC7hgps9vjZ5I0cnRilj8JYmhIaTdTfExmOlBpHgekszlXzXiH+53VTHV76GRNJqqkg00VhypGOUZEJ6jNJieZjQzCRzNyKyBBLTLRJrmxCcOdfXiSts6pr+M15pXY1i6MEh3AMp+DCBdTgGhrQBAKP8Ayv8GY9WS/Wu/UxbV2yZjMH8AfW5w+KIpiI</latexit><latexit sha1_base64="+CqQuVHMZT19nMiKQttiPx5YtbA=">AAACBnicbVBNS8NAEJ34WetX1KMIi0XwVBIR9FjsxWMV+wFNLJvtpl262YTdjVBCTl78K148KOLV3+DNf+Om7UFbHww83pthZl6QcKa043xbS8srq2vrpY3y5tb2zq69t99ScSoJbZKYx7ITYEU5E7Spmea0k0iKo4DTdjCqF377gUrFYnGnxwn1IzwQLGQEayP17KM68phAXoT1MAiy2/w+i5CnWUQVSvKeXXGqzgRokbgzUoEZGj37y+vHJI2o0IRjpbquk2g/w1Izwmle9lJFE0xGeEC7hgps9vjZ5I0cnRilj8JYmhIaTdTfExmOlBpHgekszlXzXiH+53VTHV76GRNJqqkg00VhypGOUZEJ6jNJieZjQzCRzNyKyBBLTLRJrmxCcOdfXiSts6pr+M15pXY1i6MEh3AMp+DCBdTgGhrQBAKP8Ayv8GY9WS/Wu/UxbV2yZjMH8AfW5w+KIpiI</latexit><latexit sha1_base64="+CqQuVHMZT19nMiKQttiPx5YtbA=">AAACBnicbVBNS8NAEJ34WetX1KMIi0XwVBIR9FjsxWMV+wFNLJvtpl262YTdjVBCTl78K148KOLV3+DNf+Om7UFbHww83pthZl6QcKa043xbS8srq2vrpY3y5tb2zq69t99ScSoJbZKYx7ITYEU5E7Spmea0k0iKo4DTdjCqF377gUrFYnGnxwn1IzwQLGQEayP17KM68phAXoT1MAiy2/w+i5CnWUQVSvKeXXGqzgRokbgzUoEZGj37y+vHJI2o0IRjpbquk2g/w1Izwmle9lJFE0xGeEC7hgps9vjZ5I0cnRilj8JYmhIaTdTfExmOlBpHgekszlXzXiH+53VTHV76GRNJqqkg00VhypGOUZEJ6jNJieZjQzCRzNyKyBBLTLRJrmxCcOdfXiSts6pr+M15pXY1i6MEh3AMp+DCBdTgGhrQBAKP8Ayv8GY9WS/Wu/UxbV2yZjMH8AfW5w+KIpiI</latexit><latexit sha1_base64="+CqQuVHMZT19nMiKQttiPx5YtbA=">AAACBnicbVBNS8NAEJ34WetX1KMIi0XwVBIR9FjsxWMV+wFNLJvtpl262YTdjVBCTl78K148KOLV3+DNf+Om7UFbHww83pthZl6QcKa043xbS8srq2vrpY3y5tb2zq69t99ScSoJbZKYx7ITYEU5E7Spmea0k0iKo4DTdjCqF377gUrFYnGnxwn1IzwQLGQEayP17KM68phAXoT1MAiy2/w+i5CnWUQVSvKeXXGqzgRokbgzUoEZGj37y+vHJI2o0IRjpbquk2g/w1Izwmle9lJFE0xGeEC7hgps9vjZ5I0cnRilj8JYmhIaTdTfExmOlBpHgekszlXzXiH+53VTHV76GRNJqqkg00VhypGOUZEJ6jNJieZjQzCRzNyKyBBLTLRJrmxCcOdfXiSts6pr+M15pXY1i6MEh3AMp+DCBdTgGhrQBAKP8Ayv8GY9WS/Wu/UxbV2yZjMH8AfW5w+KIpiI</latexit>

A 2 Rm⇥n<latexit sha1_base64="MxwCSL/gyMr0pbYAXqcODXgj2Pw=">AAACBnicbVBNS8NAEJ34WetX1KMIi0XwVBIR9Fj14rGK/YAmls120y7dbMLuRighJy/+FS8eFPHqb/Dmv3HT9qCtDwYe780wMy9IOFPacb6thcWl5ZXV0lp5fWNza9ve2W2qOJWENkjMY9kOsKKcCdrQTHPaTiTFUcBpKxheFX7rgUrFYnGnRwn1I9wXLGQEayN17YML5DGBvAjrQRBkt/l9FiFPs4gqJPKuXXGqzhhonrhTUoEp6l37y+vFJI2o0IRjpTquk2g/w1Izwmle9lJFE0yGuE87hgps9vjZ+I0cHRmlh8JYmhIajdXfExmOlBpFgekszlWzXiH+53VSHZ77GRNJqqkgk0VhypGOUZEJ6jFJieYjQzCRzNyKyABLTLRJrmxCcGdfnifNk6pr+M1ppXY5jaME+3AIx+DCGdTgGurQAAKP8Ayv8GY9WS/Wu/UxaV2wpjN78AfW5w+D2JiE</latexit><latexit sha1_base64="MxwCSL/gyMr0pbYAXqcODXgj2Pw=">AAACBnicbVBNS8NAEJ34WetX1KMIi0XwVBIR9Fj14rGK/YAmls120y7dbMLuRighJy/+FS8eFPHqb/Dmv3HT9qCtDwYe780wMy9IOFPacb6thcWl5ZXV0lp5fWNza9ve2W2qOJWENkjMY9kOsKKcCdrQTHPaTiTFUcBpKxheFX7rgUrFYnGnRwn1I9wXLGQEayN17YML5DGBvAjrQRBkt/l9FiFPs4gqJPKuXXGqzhhonrhTUoEp6l37y+vFJI2o0IRjpTquk2g/w1Izwmle9lJFE0yGuE87hgps9vjZ+I0cHRmlh8JYmhIajdXfExmOlBpFgekszlWzXiH+53VSHZ77GRNJqqkgk0VhypGOUZEJ6jFJieYjQzCRzNyKyABLTLRJrmxCcGdfnifNk6pr+M1ppXY5jaME+3AIx+DCGdTgGurQAAKP8Ayv8GY9WS/Wu/UxaV2wpjN78AfW5w+D2JiE</latexit><latexit sha1_base64="MxwCSL/gyMr0pbYAXqcODXgj2Pw=">AAACBnicbVBNS8NAEJ34WetX1KMIi0XwVBIR9Fj14rGK/YAmls120y7dbMLuRighJy/+FS8eFPHqb/Dmv3HT9qCtDwYe780wMy9IOFPacb6thcWl5ZXV0lp5fWNza9ve2W2qOJWENkjMY9kOsKKcCdrQTHPaTiTFUcBpKxheFX7rgUrFYnGnRwn1I9wXLGQEayN17YML5DGBvAjrQRBkt/l9FiFPs4gqJPKuXXGqzhhonrhTUoEp6l37y+vFJI2o0IRjpTquk2g/w1Izwmle9lJFE0yGuE87hgps9vjZ+I0cHRmlh8JYmhIajdXfExmOlBpFgekszlWzXiH+53VSHZ77GRNJqqkgk0VhypGOUZEJ6jFJieYjQzCRzNyKyABLTLRJrmxCcGdfnifNk6pr+M1ppXY5jaME+3AIx+DCGdTgGurQAAKP8Ayv8GY9WS/Wu/UxaV2wpjN78AfW5w+D2JiE</latexit><latexit sha1_base64="MxwCSL/gyMr0pbYAXqcODXgj2Pw=">AAACBnicbVBNS8NAEJ34WetX1KMIi0XwVBIR9Fj14rGK/YAmls120y7dbMLuRighJy/+FS8eFPHqb/Dmv3HT9qCtDwYe780wMy9IOFPacb6thcWl5ZXV0lp5fWNza9ve2W2qOJWENkjMY9kOsKKcCdrQTHPaTiTFUcBpKxheFX7rgUrFYnGnRwn1I9wXLGQEayN17YML5DGBvAjrQRBkt/l9FiFPs4gqJPKuXXGqzhhonrhTUoEp6l37y+vFJI2o0IRjpTquk2g/w1Izwmle9lJFE0yGuE87hgps9vjZ+I0cHRmlh8JYmhIajdXfExmOlBpFgekszlWzXiH+53VSHZ77GRNJqqkgk0VhypGOUZEJ6jFJieYjQzCRzNyKyABLTLRJrmxCcGdfnifNk6pr+M1ppXY5jaME+3AIx+DCGdTgGurQAAKP8Ayv8GY9WS/Wu/UxaV2wpjN78AfW5w+D2JiE</latexit>

B 2 Rn⇥p<latexit sha1_base64="EXB3w9wWi16rdVHdgvQBZlzMJFo=">AAACBnicbVDLSsNAFL3xWesr6lKEwSK4KokIuix147KKfUATy2Q6aYdOJmFmIpSQlRt/xY0LRdz6De78GydtF9p6YODMOfdy7z1BwpnSjvNtLS2vrK6tlzbKm1vbO7v23n5LxakktEliHstOgBXlTNCmZprTTiIpjgJO28HoqvDbD1QqFos7PU6oH+GBYCEjWBupZx/VkccE8iKsh0GQ3eb3mflpFlGFkrxnV5yqMwFaJO6MVGCGRs/+8voxSSMqNOFYqa7rJNrPsNSMcJqXvVTRBJMRHtCuoQKbOX42OSNHJ0bpozCW5gmNJurvjgxHSo2jwFQW66p5rxD/87qpDi/9jIkk1VSQ6aAw5UjHqMgE9ZmkRPOxIZhIZnZFZIglJtokVzYhuPMnL5LWWdU1/Oa8UqvP4ijBIRzDKbhwATW4hgY0gcAjPMMrvFlP1ov1bn1MS5esWc8B/IH1+QOKEJiI</latexit><latexit sha1_base64="EXB3w9wWi16rdVHdgvQBZlzMJFo=">AAACBnicbVDLSsNAFL3xWesr6lKEwSK4KokIuix147KKfUATy2Q6aYdOJmFmIpSQlRt/xY0LRdz6De78GydtF9p6YODMOfdy7z1BwpnSjvNtLS2vrK6tlzbKm1vbO7v23n5LxakktEliHstOgBXlTNCmZprTTiIpjgJO28HoqvDbD1QqFos7PU6oH+GBYCEjWBupZx/VkccE8iKsh0GQ3eb3mflpFlGFkrxnV5yqMwFaJO6MVGCGRs/+8voxSSMqNOFYqa7rJNrPsNSMcJqXvVTRBJMRHtCuoQKbOX42OSNHJ0bpozCW5gmNJurvjgxHSo2jwFQW66p5rxD/87qpDi/9jIkk1VSQ6aAw5UjHqMgE9ZmkRPOxIZhIZnZFZIglJtokVzYhuPMnL5LWWdU1/Oa8UqvP4ijBIRzDKbhwATW4hgY0gcAjPMMrvFlP1ov1bn1MS5esWc8B/IH1+QOKEJiI</latexit><latexit sha1_base64="EXB3w9wWi16rdVHdgvQBZlzMJFo=">AAACBnicbVDLSsNAFL3xWesr6lKEwSK4KokIuix147KKfUATy2Q6aYdOJmFmIpSQlRt/xY0LRdz6De78GydtF9p6YODMOfdy7z1BwpnSjvNtLS2vrK6tlzbKm1vbO7v23n5LxakktEliHstOgBXlTNCmZprTTiIpjgJO28HoqvDbD1QqFos7PU6oH+GBYCEjWBupZx/VkccE8iKsh0GQ3eb3mflpFlGFkrxnV5yqMwFaJO6MVGCGRs/+8voxSSMqNOFYqa7rJNrPsNSMcJqXvVTRBJMRHtCuoQKbOX42OSNHJ0bpozCW5gmNJurvjgxHSo2jwFQW66p5rxD/87qpDi/9jIkk1VSQ6aAw5UjHqMgE9ZmkRPOxIZhIZnZFZIglJtokVzYhuPMnL5LWWdU1/Oa8UqvP4ijBIRzDKbhwATW4hgY0gcAjPMMrvFlP1ov1bn1MS5esWc8B/IH1+QOKEJiI</latexit><latexit sha1_base64="EXB3w9wWi16rdVHdgvQBZlzMJFo=">AAACBnicbVDLSsNAFL3xWesr6lKEwSK4KokIuix147KKfUATy2Q6aYdOJmFmIpSQlRt/xY0LRdz6De78GydtF9p6YODMOfdy7z1BwpnSjvNtLS2vrK6tlzbKm1vbO7v23n5LxakktEliHstOgBXlTNCmZprTTiIpjgJO28HoqvDbD1QqFos7PU6oH+GBYCEjWBupZx/VkccE8iKsh0GQ3eb3mflpFlGFkrxnV5yqMwFaJO6MVGCGRs/+8voxSSMqNOFYqa7rJNrPsNSMcJqXvVTRBJMRHtCuoQKbOX42OSNHJ0bpozCW5gmNJurvjgxHSo2jwFQW66p5rxD/87qpDi/9jIkk1VSQ6aAw5UjHqMgE9ZmkRPOxIZhIZnZFZIglJtokVzYhuPMnL5LWWdU1/Oa8UqvP4ijBIRzDKbhwATW4hgY0gcAjPMMrvFlP1ov1bn1MS5esWc8B/IH1+QOKEJiI</latexit>

- Properties:(AB)C = A(BC), 8A,B,C

↵(AB) = (↵A)B, 8A,B

A(B + C) = AB +AC, 8A,B,C

(AB)> = B>A>, 8, A,BAB 6= BA

<latexit sha1_base64="O0FDJ9gN5tsRKG8Zzmu2iR0fabg=">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</latexit><latexit sha1_base64="O0FDJ9gN5tsRKG8Zzmu2iR0fabg=">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</latexit><latexit sha1_base64="O0FDJ9gN5tsRKG8Zzmu2iR0fabg=">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</latexit><latexit sha1_base64="O0FDJ9gN5tsRKG8Zzmu2iR0fabg=">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</latexit>

<latexit sha1_base64="hfRm/dkx6hM/0KpRtYEuroYj530=">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</latexit>2

6664

C11 C12 · · · C1p

C21 C22 · · · C2p...

. . ....

Cm1 Cm2 · · · Cmp

3

7775= C = AB =

2

6664

A11 A12 · · · A1n

A21 A22 · · · A2n...

. . ....

Am1 Am2 · · · Amn

3

7775·

2

6664

B11 B12 · · · B1p

B21 B22 · · · B2p...

. . ....

Bn1 Bn2 · · · Bnp

3

7775

Page 124: COMP 414/514: Optimization - Algorithms, Complexity and

Matrices- Inner product:

hA,Bi = Tr(A>B) = Tr(B>A), 8A,B 2 Rm⇥n<latexit sha1_base64="8hSszzx+smyLVbIvoGTxVRLNqUw=">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</latexit><latexit sha1_base64="8hSszzx+smyLVbIvoGTxVRLNqUw=">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</latexit><latexit sha1_base64="8hSszzx+smyLVbIvoGTxVRLNqUw=">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</latexit><latexit sha1_base64="8hSszzx+smyLVbIvoGTxVRLNqUw=">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</latexit>

Page 125: COMP 414/514: Optimization - Algorithms, Complexity and

Matrices- Inner product:

hA,Bi = Tr(A>B) = Tr(B>A), 8A,B 2 Rm⇥n<latexit sha1_base64="8hSszzx+smyLVbIvoGTxVRLNqUw=">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</latexit><latexit sha1_base64="8hSszzx+smyLVbIvoGTxVRLNqUw=">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</latexit><latexit sha1_base64="8hSszzx+smyLVbIvoGTxVRLNqUw=">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</latexit><latexit sha1_base64="8hSszzx+smyLVbIvoGTxVRLNqUw=">AAACYHicbZFNTxsxEIa9Cy1pgBLaW3sZEVUCCaHdqhK9VCLhwhEqAkhxiLzObGLh9a7s2bbRav9kbz1w6S/BSfbA10iWXj0z4xm/TgqtHEXRvyBcW3/zdqP1rr25tf1+p7P74crlpZU4kLnO7U0iHGplcECKNN4UFkWWaLxO7k4X+etfaJ3KzSXNCxxlYmpUqqQgj8ad31xjSsC1MFON0DuEPnCrpjPP7Ir9AE74h6pLW+/3bjnlBfQPntD+ivYODoGnuRVaNxcpAzwTNEuS6md9W2W+R2XowNTjTjc6ipYBL0XciC5r4nzc+csnuSwzNCS1cG4YRwWNKmFJSY11m5cOCyHvxBSHXhrh54yqpUE1fPFkAn41fwzBkj7uqETm3DxLfOViXfc8t4Cv5YYlpd9HlTJFSWjkalBaaqAcFm7DRFmUpOdeCGmV3xXkTFghyf9J25sQP3/yS3H19Sj2+uJb96Tf2NFin9ke22cxO2Yn7IydswGT7D5YC7aC7eB/2Ap3wt1VaRg0PR/Zkwg/PQAPIrMT</latexit>

- Example:

<latexit sha1_base64="08VdpfPHvDN0QsxN3EeHOP7Epb4=">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</latexit>

A =

2

6664

A11 A12 · · · A1n

A21 A22 · · · A2n...

. . ....

Am1 Am2 · · · Amn

3

7775

<latexit sha1_base64="+USZw/u70jAQB8GoO/8YyMgtnx0=">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</latexit>

B =

2

6664

B11 B12 · · · B1n

B21 B22 · · · B2n...

. . ....

Bm1 Bn2 · · · Bmn

3

7775

Page 126: COMP 414/514: Optimization - Algorithms, Complexity and

Matrices- Inner product:

hA,Bi = Tr(A>B) = Tr(B>A), 8A,B 2 Rm⇥n<latexit sha1_base64="8hSszzx+smyLVbIvoGTxVRLNqUw=">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</latexit><latexit sha1_base64="8hSszzx+smyLVbIvoGTxVRLNqUw=">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</latexit><latexit sha1_base64="8hSszzx+smyLVbIvoGTxVRLNqUw=">AAACYHicbZFNTxsxEIa9Cy1pgBLaW3sZEVUCCaHdqhK9VCLhwhEqAkhxiLzObGLh9a7s2bbRav9kbz1w6S/BSfbA10iWXj0z4xm/TgqtHEXRvyBcW3/zdqP1rr25tf1+p7P74crlpZU4kLnO7U0iHGplcECKNN4UFkWWaLxO7k4X+etfaJ3KzSXNCxxlYmpUqqQgj8ad31xjSsC1MFON0DuEPnCrpjPP7Ir9AE74h6pLW+/3bjnlBfQPntD+ivYODoGnuRVaNxcpAzwTNEuS6md9W2W+R2XowNTjTjc6ipYBL0XciC5r4nzc+csnuSwzNCS1cG4YRwWNKmFJSY11m5cOCyHvxBSHXhrh54yqpUE1fPFkAn41fwzBkj7uqETm3DxLfOViXfc8t4Cv5YYlpd9HlTJFSWjkalBaaqAcFm7DRFmUpOdeCGmV3xXkTFghyf9J25sQP3/yS3H19Sj2+uJb96Tf2NFin9ke22cxO2Yn7IydswGT7D5YC7aC7eB/2Ap3wt1VaRg0PR/Zkwg/PQAPIrMT</latexit><latexit sha1_base64="8hSszzx+smyLVbIvoGTxVRLNqUw=">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</latexit>

- Example:

<latexit sha1_base64="08VdpfPHvDN0QsxN3EeHOP7Epb4=">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</latexit>

A =

2

6664

A11 A12 · · · A1n

A21 A22 · · · A2n...

. . ....

Am1 Am2 · · · Amn

3

7775

<latexit sha1_base64="+USZw/u70jAQB8GoO/8YyMgtnx0=">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</latexit>

B =

2

6664

B11 B12 · · · B1n

B21 B22 · · · B2n...

. . ....

Bm1 Bn2 · · · Bmn

3

7775

<latexit sha1_base64="dZ6E/YMVhgbeC6MuVL2EO/dqSZs=">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</latexit>

A>B =

2

6664

Pmi=1 Ai1 ·Bi1 · · · · · · · · ·

· · ·Pm

i=1 Ai2 ·Bi2 · · · · · ·...

. . ....

· · · · · · · · ·Pm

i=1 Ain ·Bin

3

7775

Page 127: COMP 414/514: Optimization - Algorithms, Complexity and

Matrices- Inner product:

hA,Bi = Tr(A>B) = Tr(B>A), 8A,B 2 Rm⇥n<latexit sha1_base64="8hSszzx+smyLVbIvoGTxVRLNqUw=">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</latexit><latexit sha1_base64="8hSszzx+smyLVbIvoGTxVRLNqUw=">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</latexit><latexit sha1_base64="8hSszzx+smyLVbIvoGTxVRLNqUw=">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</latexit><latexit sha1_base64="8hSszzx+smyLVbIvoGTxVRLNqUw=">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</latexit>

- Example:

<latexit sha1_base64="08VdpfPHvDN0QsxN3EeHOP7Epb4=">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</latexit>

A =

2

6664

A11 A12 · · · A1n

A21 A22 · · · A2n...

. . ....

Am1 Am2 · · · Amn

3

7775

<latexit sha1_base64="+USZw/u70jAQB8GoO/8YyMgtnx0=">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</latexit>

B =

2

6664

B11 B12 · · · B1n

B21 B22 · · · B2n...

. . ....

Bm1 Bn2 · · · Bmn

3

7775

<latexit sha1_base64="fObuId0f5goXS4Wk9l8M7T32cHo=">AAACc3icbZFLSwMxEMez66vWV1Xw4sFgFRSx7BZFL0KtF48KVoVuLdk0rcE8lmRWLMt+AT+eN7+FF++mj0PVDgT+85/fkMwkTgS3EASfnj8zOze/UFgsLi2vrK6V1jfurU4NZQ2qhTaPMbFMcMUawEGwx8QwImPBHuKXq0H94ZUZy7W6g37CWpL0FO9ySsBZ7dJ7BOwNsjuTH1w+RaATXD/EFziyqWxn3Kkwf5L40ukwxxHtaMD1UXI0DapOQtUhNEgtPpoCq0lY5e1SOagEw8D/RTgWZTSOm3bpI+pomkqmgApibTMMEmhlxACnguXFKLUsIfSF9FjTSUUks61suLMc7zung7vauKMAD93JjoxIa/sydqQk8Gz/1gbmtFozhe55y82TpMAUHV3UTQUGjQcfgDvcMAqi7wShhru3YvpMDKHgvqnolhD+Hfm/uK9WwtNKcHtSrtXH6yigbbSLDlCIzlANXaMb1EAUfXlb3o6HvW9/29/190ao7417NtGv8I9/ANNPuaw=</latexit>

Tr(A>B) =mX

i=1

Ai1 ·Bi1 +mX

i=1

Ai2 ·Bi2 + · · ·+mX

i=1

Ain ·Bin

<latexit sha1_base64="dZ6E/YMVhgbeC6MuVL2EO/dqSZs=">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</latexit>

A>B =

2

6664

Pmi=1 Ai1 ·Bi1 · · · · · · · · ·

· · ·Pm

i=1 Ai2 ·Bi2 · · · · · ·...

. . ....

· · · · · · · · ·Pm

i=1 Ain ·Bin

3

7775

Page 128: COMP 414/514: Optimization - Algorithms, Complexity and

Matrices- Inner product:

hA,Bi = Tr(A>B) = Tr(B>A), 8A,B 2 Rm⇥n<latexit sha1_base64="8hSszzx+smyLVbIvoGTxVRLNqUw=">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</latexit><latexit sha1_base64="8hSszzx+smyLVbIvoGTxVRLNqUw=">AAACYHicbZFNTxsxEIa9Cy1pgBLaW3sZEVUCCaHdqhK9VCLhwhEqAkhxiLzObGLh9a7s2bbRav9kbz1w6S/BSfbA10iWXj0z4xm/TgqtHEXRvyBcW3/zdqP1rr25tf1+p7P74crlpZU4kLnO7U0iHGplcECKNN4UFkWWaLxO7k4X+etfaJ3KzSXNCxxlYmpUqqQgj8ad31xjSsC1MFON0DuEPnCrpjPP7Ir9AE74h6pLW+/3bjnlBfQPntD+ivYODoGnuRVaNxcpAzwTNEuS6md9W2W+R2XowNTjTjc6ipYBL0XciC5r4nzc+csnuSwzNCS1cG4YRwWNKmFJSY11m5cOCyHvxBSHXhrh54yqpUE1fPFkAn41fwzBkj7uqETm3DxLfOViXfc8t4Cv5YYlpd9HlTJFSWjkalBaaqAcFm7DRFmUpOdeCGmV3xXkTFghyf9J25sQP3/yS3H19Sj2+uJb96Tf2NFin9ke22cxO2Yn7IydswGT7D5YC7aC7eB/2Ap3wt1VaRg0PR/Zkwg/PQAPIrMT</latexit><latexit sha1_base64="8hSszzx+smyLVbIvoGTxVRLNqUw=">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</latexit><latexit sha1_base64="8hSszzx+smyLVbIvoGTxVRLNqUw=">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</latexit>

- Example:

<latexit sha1_base64="08VdpfPHvDN0QsxN3EeHOP7Epb4=">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</latexit>

A =

2

6664

A11 A12 · · · A1n

A21 A22 · · · A2n...

. . ....

Am1 Am2 · · · Amn

3

7775

<latexit sha1_base64="+USZw/u70jAQB8GoO/8YyMgtnx0=">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</latexit>

B =

2

6664

B11 B12 · · · B1n

B21 B22 · · · B2n...

. . ....

Bm1 Bn2 · · · Bmn

3

7775

<latexit sha1_base64="dZ6E/YMVhgbeC6MuVL2EO/dqSZs=">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</latexit>

A>B =

2

6664

Pmi=1 Ai1 ·Bi1 · · · · · · · · ·

· · ·Pm

i=1 Ai2 ·Bi2 · · · · · ·...

. . ....

· · · · · · · · ·Pm

i=1 Ain ·Bin

3

7775

<latexit sha1_base64="3WgZ9LFNXoLlBeQDfRhSpkRk9rI=">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</latexit>

Tr(A>B) = vec(A)>vec(B) = hvec(A), vec(B)i

Page 129: COMP 414/514: Optimization - Algorithms, Complexity and

Matrices- Inner product:

hA,Bi = Tr(A>B) = Tr(B>A), 8A,B 2 Rm⇥n<latexit sha1_base64="8hSszzx+smyLVbIvoGTxVRLNqUw=">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</latexit><latexit sha1_base64="8hSszzx+smyLVbIvoGTxVRLNqUw=">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</latexit><latexit sha1_base64="8hSszzx+smyLVbIvoGTxVRLNqUw=">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</latexit><latexit sha1_base64="8hSszzx+smyLVbIvoGTxVRLNqUw=">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</latexit>

Page 130: COMP 414/514: Optimization - Algorithms, Complexity and

Matrices- Inner product:

hA,Bi = Tr(A>B) = Tr(B>A), 8A,B 2 Rm⇥n<latexit sha1_base64="8hSszzx+smyLVbIvoGTxVRLNqUw=">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</latexit><latexit sha1_base64="8hSszzx+smyLVbIvoGTxVRLNqUw=">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</latexit><latexit sha1_base64="8hSszzx+smyLVbIvoGTxVRLNqUw=">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</latexit><latexit sha1_base64="8hSszzx+smyLVbIvoGTxVRLNqUw=">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</latexit>

- Rank of a matrix: maximum # of independent columns or rows

Page 131: COMP 414/514: Optimization - Algorithms, Complexity and

Matrices- Inner product:

hA,Bi = Tr(A>B) = Tr(B>A), 8A,B 2 Rm⇥n<latexit sha1_base64="8hSszzx+smyLVbIvoGTxVRLNqUw=">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</latexit><latexit sha1_base64="8hSszzx+smyLVbIvoGTxVRLNqUw=">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</latexit><latexit sha1_base64="8hSszzx+smyLVbIvoGTxVRLNqUw=">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</latexit><latexit sha1_base64="8hSszzx+smyLVbIvoGTxVRLNqUw=">AAACYHicbZFNTxsxEIa9Cy1pgBLaW3sZEVUCCaHdqhK9VCLhwhEqAkhxiLzObGLh9a7s2bbRav9kbz1w6S/BSfbA10iWXj0z4xm/TgqtHEXRvyBcW3/zdqP1rr25tf1+p7P74crlpZU4kLnO7U0iHGplcECKNN4UFkWWaLxO7k4X+etfaJ3KzSXNCxxlYmpUqqQgj8ad31xjSsC1MFON0DuEPnCrpjPP7Ir9AE74h6pLW+/3bjnlBfQPntD+ivYODoGnuRVaNxcpAzwTNEuS6md9W2W+R2XowNTjTjc6ipYBL0XciC5r4nzc+csnuSwzNCS1cG4YRwWNKmFJSY11m5cOCyHvxBSHXhrh54yqpUE1fPFkAn41fwzBkj7uqETm3DxLfOViXfc8t4Cv5YYlpd9HlTJFSWjkalBaaqAcFm7DRFmUpOdeCGmV3xXkTFghyf9J25sQP3/yS3H19Sj2+uJb96Tf2NFin9ke22cxO2Yn7IydswGT7D5YC7aC7eB/2Ap3wt1VaRg0PR/Zkwg/PQAPIrMT</latexit>

- Rank of a matrix: maximum # of independent columns or rows

- Nullspace of a matrix: {x | Ax = 0}<latexit sha1_base64="ETKq9O6nCwqPcIWtxaE+4CU7rNg=">AAAB+XicbZDLSsNAFIZPvNZ6i7p0M1gEVyURQTdC1Y3LCvYCTSiT6aQdOpmEmUlpie2TuHGhiFvfxJ1v47TNQlt/GPj4zzmcM3+QcKa043xbK6tr6xubha3i9s7u3r59cFhXcSoJrZGYx7IZYEU5E7Smmea0mUiKo4DTRtC/m9YbAyoVi8WjHiXUj3BXsJARrI3Vtm0vG6LJ0wTdDNE1crxx2y45ZWcmtAxuDiXIVW3bX14nJmlEhSYcK9VynUT7GZaaEU7HRS9VNMGkj7u0ZVDgiCo/m10+RqfG6aAwluYJjWbu74kMR0qNosB0Rlj31GJtav5Xa6U6vPIzJpJUU0Hmi8KUIx2jaQyowyQlmo8MYCKZuRWRHpaYaBNW0YTgLn55GernZdfww0WpcpvHUYBjOIEzcOESKnAPVagBgQE8wyu8WZn1Yr1bH/PWFSufOYI/sj5/APu4kpE=</latexit><latexit sha1_base64="ETKq9O6nCwqPcIWtxaE+4CU7rNg=">AAAB+XicbZDLSsNAFIZPvNZ6i7p0M1gEVyURQTdC1Y3LCvYCTSiT6aQdOpmEmUlpie2TuHGhiFvfxJ1v47TNQlt/GPj4zzmcM3+QcKa043xbK6tr6xubha3i9s7u3r59cFhXcSoJrZGYx7IZYEU5E7Smmea0mUiKo4DTRtC/m9YbAyoVi8WjHiXUj3BXsJARrI3Vtm0vG6LJ0wTdDNE1crxx2y45ZWcmtAxuDiXIVW3bX14nJmlEhSYcK9VynUT7GZaaEU7HRS9VNMGkj7u0ZVDgiCo/m10+RqfG6aAwluYJjWbu74kMR0qNosB0Rlj31GJtav5Xa6U6vPIzJpJUU0Hmi8KUIx2jaQyowyQlmo8MYCKZuRWRHpaYaBNW0YTgLn55GernZdfww0WpcpvHUYBjOIEzcOESKnAPVagBgQE8wyu8WZn1Yr1bH/PWFSufOYI/sj5/APu4kpE=</latexit><latexit sha1_base64="ETKq9O6nCwqPcIWtxaE+4CU7rNg=">AAAB+XicbZDLSsNAFIZPvNZ6i7p0M1gEVyURQTdC1Y3LCvYCTSiT6aQdOpmEmUlpie2TuHGhiFvfxJ1v47TNQlt/GPj4zzmcM3+QcKa043xbK6tr6xubha3i9s7u3r59cFhXcSoJrZGYx7IZYEU5E7Smmea0mUiKo4DTRtC/m9YbAyoVi8WjHiXUj3BXsJARrI3Vtm0vG6LJ0wTdDNE1crxx2y45ZWcmtAxuDiXIVW3bX14nJmlEhSYcK9VynUT7GZaaEU7HRS9VNMGkj7u0ZVDgiCo/m10+RqfG6aAwluYJjWbu74kMR0qNosB0Rlj31GJtav5Xa6U6vPIzJpJUU0Hmi8KUIx2jaQyowyQlmo8MYCKZuRWRHpaYaBNW0YTgLn55GernZdfww0WpcpvHUYBjOIEzcOESKnAPVagBgQE8wyu8WZn1Yr1bH/PWFSufOYI/sj5/APu4kpE=</latexit><latexit sha1_base64="ETKq9O6nCwqPcIWtxaE+4CU7rNg=">AAAB+XicbZDLSsNAFIZPvNZ6i7p0M1gEVyURQTdC1Y3LCvYCTSiT6aQdOpmEmUlpie2TuHGhiFvfxJ1v47TNQlt/GPj4zzmcM3+QcKa043xbK6tr6xubha3i9s7u3r59cFhXcSoJrZGYx7IZYEU5E7Smmea0mUiKo4DTRtC/m9YbAyoVi8WjHiXUj3BXsJARrI3Vtm0vG6LJ0wTdDNE1crxx2y45ZWcmtAxuDiXIVW3bX14nJmlEhSYcK9VynUT7GZaaEU7HRS9VNMGkj7u0ZVDgiCo/m10+RqfG6aAwluYJjWbu74kMR0qNosB0Rlj31GJtav5Xa6U6vPIzJpJUU0Hmi8KUIx2jaQyowyQlmo8MYCKZuRWRHpaYaBNW0YTgLn55GernZdfww0WpcpvHUYBjOIEzcOESKnAPVagBgQE8wyu8WZn1Yr1bH/PWFSufOYI/sj5/APu4kpE=</latexit>

Page 132: COMP 414/514: Optimization - Algorithms, Complexity and

Matrices- Inner product:

hA,Bi = Tr(A>B) = Tr(B>A), 8A,B 2 Rm⇥n<latexit sha1_base64="8hSszzx+smyLVbIvoGTxVRLNqUw=">AAACYHicbZFNTxsxEIa9Cy1pgBLaW3sZEVUCCaHdqhK9VCLhwhEqAkhxiLzObGLh9a7s2bbRav9kbz1w6S/BSfbA10iWXj0z4xm/TgqtHEXRvyBcW3/zdqP1rr25tf1+p7P74crlpZU4kLnO7U0iHGplcECKNN4UFkWWaLxO7k4X+etfaJ3KzSXNCxxlYmpUqqQgj8ad31xjSsC1MFON0DuEPnCrpjPP7Ir9AE74h6pLW+/3bjnlBfQPntD+ivYODoGnuRVaNxcpAzwTNEuS6md9W2W+R2XowNTjTjc6ipYBL0XciC5r4nzc+csnuSwzNCS1cG4YRwWNKmFJSY11m5cOCyHvxBSHXhrh54yqpUE1fPFkAn41fwzBkj7uqETm3DxLfOViXfc8t4Cv5YYlpd9HlTJFSWjkalBaaqAcFm7DRFmUpOdeCGmV3xXkTFghyf9J25sQP3/yS3H19Sj2+uJb96Tf2NFin9ke22cxO2Yn7IydswGT7D5YC7aC7eB/2Ap3wt1VaRg0PR/Zkwg/PQAPIrMT</latexit><latexit sha1_base64="8hSszzx+smyLVbIvoGTxVRLNqUw=">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</latexit><latexit sha1_base64="8hSszzx+smyLVbIvoGTxVRLNqUw=">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</latexit><latexit sha1_base64="8hSszzx+smyLVbIvoGTxVRLNqUw=">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</latexit>

- Rank of a matrix: maximum # of independent columns or rows

- Nullspace of a matrix: {x | Ax = 0}<latexit sha1_base64="ETKq9O6nCwqPcIWtxaE+4CU7rNg=">AAAB+XicbZDLSsNAFIZPvNZ6i7p0M1gEVyURQTdC1Y3LCvYCTSiT6aQdOpmEmUlpie2TuHGhiFvfxJ1v47TNQlt/GPj4zzmcM3+QcKa043xbK6tr6xubha3i9s7u3r59cFhXcSoJrZGYx7IZYEU5E7Smmea0mUiKo4DTRtC/m9YbAyoVi8WjHiXUj3BXsJARrI3Vtm0vG6LJ0wTdDNE1crxx2y45ZWcmtAxuDiXIVW3bX14nJmlEhSYcK9VynUT7GZaaEU7HRS9VNMGkj7u0ZVDgiCo/m10+RqfG6aAwluYJjWbu74kMR0qNosB0Rlj31GJtav5Xa6U6vPIzJpJUU0Hmi8KUIx2jaQyowyQlmo8MYCKZuRWRHpaYaBNW0YTgLn55GernZdfww0WpcpvHUYBjOIEzcOESKnAPVagBgQE8wyu8WZn1Yr1bH/PWFSufOYI/sj5/APu4kpE=</latexit><latexit sha1_base64="ETKq9O6nCwqPcIWtxaE+4CU7rNg=">AAAB+XicbZDLSsNAFIZPvNZ6i7p0M1gEVyURQTdC1Y3LCvYCTSiT6aQdOpmEmUlpie2TuHGhiFvfxJ1v47TNQlt/GPj4zzmcM3+QcKa043xbK6tr6xubha3i9s7u3r59cFhXcSoJrZGYx7IZYEU5E7Smmea0mUiKo4DTRtC/m9YbAyoVi8WjHiXUj3BXsJARrI3Vtm0vG6LJ0wTdDNE1crxx2y45ZWcmtAxuDiXIVW3bX14nJmlEhSYcK9VynUT7GZaaEU7HRS9VNMGkj7u0ZVDgiCo/m10+RqfG6aAwluYJjWbu74kMR0qNosB0Rlj31GJtav5Xa6U6vPIzJpJUU0Hmi8KUIx2jaQyowyQlmo8MYCKZuRWRHpaYaBNW0YTgLn55GernZdfww0WpcpvHUYBjOIEzcOESKnAPVagBgQE8wyu8WZn1Yr1bH/PWFSufOYI/sj5/APu4kpE=</latexit><latexit sha1_base64="ETKq9O6nCwqPcIWtxaE+4CU7rNg=">AAAB+XicbZDLSsNAFIZPvNZ6i7p0M1gEVyURQTdC1Y3LCvYCTSiT6aQdOpmEmUlpie2TuHGhiFvfxJ1v47TNQlt/GPj4zzmcM3+QcKa043xbK6tr6xubha3i9s7u3r59cFhXcSoJrZGYx7IZYEU5E7Smmea0mUiKo4DTRtC/m9YbAyoVi8WjHiXUj3BXsJARrI3Vtm0vG6LJ0wTdDNE1crxx2y45ZWcmtAxuDiXIVW3bX14nJmlEhSYcK9VynUT7GZaaEU7HRS9VNMGkj7u0ZVDgiCo/m10+RqfG6aAwluYJjWbu74kMR0qNosB0Rlj31GJtav5Xa6U6vPIzJpJUU0Hmi8KUIx2jaQyowyQlmo8MYCKZuRWRHpaYaBNW0YTgLn55GernZdfww0WpcpvHUYBjOIEzcOESKnAPVagBgQE8wyu8WZn1Yr1bH/PWFSufOYI/sj5/APu4kpE=</latexit><latexit sha1_base64="ETKq9O6nCwqPcIWtxaE+4CU7rNg=">AAAB+XicbZDLSsNAFIZPvNZ6i7p0M1gEVyURQTdC1Y3LCvYCTSiT6aQdOpmEmUlpie2TuHGhiFvfxJ1v47TNQlt/GPj4zzmcM3+QcKa043xbK6tr6xubha3i9s7u3r59cFhXcSoJrZGYx7IZYEU5E7Smmea0mUiKo4DTRtC/m9YbAyoVi8WjHiXUj3BXsJARrI3Vtm0vG6LJ0wTdDNE1crxx2y45ZWcmtAxuDiXIVW3bX14nJmlEhSYcK9VynUT7GZaaEU7HRS9VNMGkj7u0ZVDgiCo/m10+RqfG6aAwluYJjWbu74kMR0qNosB0Rlj31GJtav5Xa6U6vPIzJpJUU0Hmi8KUIx2jaQyowyQlmo8MYCKZuRWRHpaYaBNW0YTgLn55GernZdfww0WpcpvHUYBjOIEzcOESKnAPVagBgQE8wyu8WZn1Yr1bH/PWFSufOYI/sj5/APu4kpE=</latexit>

1. A 2 Rn⇥n

2. A is symmetric

3. x>Ax � 0, 8x 2 Rn, x 6= 0<latexit sha1_base64="etQJaxVcfVggDpD+GY20/cJ/wkM=">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</latexit><latexit sha1_base64="etQJaxVcfVggDpD+GY20/cJ/wkM=">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</latexit><latexit sha1_base64="etQJaxVcfVggDpD+GY20/cJ/wkM=">AAACbHicbVFNTxsxEPUutIX0g7TlAEKVRo2KekDRLkUqRz4uHAE1gBSHyOtMgoXXu7VnUaJVuPQfcuMncOE34A174KMjWXp+740885zkWjmKotsgnJt/8/bdwmLj/YePn5aan7+cuKywEjsy05k9S4RDrQx2SJHGs9yiSBONp8nlfqWfXqF1KjN/aJJjLxUjo4ZKCvJUv/kvbsP69S5wZYCngi6SpDyenpf+RipFB2YKnDc2axfhmEpQDtwkTZGskjP5VyWPzzllOezCGPgI/0K0Add8mFmhdUU9f8B40ZOm8vWbragdzQpeg7gGLVbXYb95wweZLFI0JLVwrhtHOfVKYUlJjdMGLxzmQl6KEXY9NMIv0itnYU3hh2cG4OfyxxDM2KcdpUid3y7xzmpc91KryP9p3YKG271SmbwgNPLxoWGhgTKokoeBsihJTzwQ0io/K8gLYYUk/z8NH0L8cuXX4GSzHXt8tNXa2avjWGBr7Dv7yWL2m+2wA3bIOkyyu2ApWAlWg/twOVwLvz1aw6Du+cqeVbj+ACnmtpU=</latexit><latexit sha1_base64="etQJaxVcfVggDpD+GY20/cJ/wkM=">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</latexit>

- Positive semi-definite matrices: A ⌫ 0<latexit sha1_base64="ukrBuXjeRn4vzfAcLDRgTtP2vYo=">AAAB8nicbZBNS8NAEIY39avWr6pHL4tF8FQSEfRY9eKxgq2FNJTNdtIu3Wzi7kQooT/DiwdFvPprvPlv3LY5aOsLCw/vzLAzb5hKYdB1v53Syura+kZ5s7K1vbO7V90/aJsk0xxaPJGJ7oTMgBQKWihQQifVwOJQwkM4upnWH55AG5GoexynEMRsoEQkOENr+Ve0azLO4ZG6vWrNrbsz0WXwCqiRQs1e9avbT3gWg0IumTG+56YY5Eyj4BImlW5mIGV8xAbgW1QsBhPks5Un9MQ6fRol2j6FdOb+nshZbMw4Dm1nzHBoFmtT87+an2F0GeRCpRmC4vOPokxSTOj0ftoXGjjKsQXGtbC7Uj5kmnG0KVVsCN7iycvQPqt7lu/Oa43rIo4yOSLH5JR45II0yC1pkhbhJCHP5JW8Oei8OO/Ox7y15BQzh+SPnM8fG+OQeQ==</latexit><latexit sha1_base64="ukrBuXjeRn4vzfAcLDRgTtP2vYo=">AAAB8nicbZBNS8NAEIY39avWr6pHL4tF8FQSEfRY9eKxgq2FNJTNdtIu3Wzi7kQooT/DiwdFvPprvPlv3LY5aOsLCw/vzLAzb5hKYdB1v53Syura+kZ5s7K1vbO7V90/aJsk0xxaPJGJ7oTMgBQKWihQQifVwOJQwkM4upnWH55AG5GoexynEMRsoEQkOENr+Ve0azLO4ZG6vWrNrbsz0WXwCqiRQs1e9avbT3gWg0IumTG+56YY5Eyj4BImlW5mIGV8xAbgW1QsBhPks5Un9MQ6fRol2j6FdOb+nshZbMw4Dm1nzHBoFmtT87+an2F0GeRCpRmC4vOPokxSTOj0ftoXGjjKsQXGtbC7Uj5kmnG0KVVsCN7iycvQPqt7lu/Oa43rIo4yOSLH5JR45II0yC1pkhbhJCHP5JW8Oei8OO/Ox7y15BQzh+SPnM8fG+OQeQ==</latexit><latexit sha1_base64="ukrBuXjeRn4vzfAcLDRgTtP2vYo=">AAAB8nicbZBNS8NAEIY39avWr6pHL4tF8FQSEfRY9eKxgq2FNJTNdtIu3Wzi7kQooT/DiwdFvPprvPlv3LY5aOsLCw/vzLAzb5hKYdB1v53Syura+kZ5s7K1vbO7V90/aJsk0xxaPJGJ7oTMgBQKWihQQifVwOJQwkM4upnWH55AG5GoexynEMRsoEQkOENr+Ve0azLO4ZG6vWrNrbsz0WXwCqiRQs1e9avbT3gWg0IumTG+56YY5Eyj4BImlW5mIGV8xAbgW1QsBhPks5Un9MQ6fRol2j6FdOb+nshZbMw4Dm1nzHBoFmtT87+an2F0GeRCpRmC4vOPokxSTOj0ftoXGjjKsQXGtbC7Uj5kmnG0KVVsCN7iycvQPqt7lu/Oa43rIo4yOSLH5JR45II0yC1pkhbhJCHP5JW8Oei8OO/Ox7y15BQzh+SPnM8fG+OQeQ==</latexit><latexit sha1_base64="ukrBuXjeRn4vzfAcLDRgTtP2vYo=">AAAB8nicbZBNS8NAEIY39avWr6pHL4tF8FQSEfRY9eKxgq2FNJTNdtIu3Wzi7kQooT/DiwdFvPprvPlv3LY5aOsLCw/vzLAzb5hKYdB1v53Syura+kZ5s7K1vbO7V90/aJsk0xxaPJGJ7oTMgBQKWihQQifVwOJQwkM4upnWH55AG5GoexynEMRsoEQkOENr+Ve0azLO4ZG6vWrNrbsz0WXwCqiRQs1e9avbT3gWg0IumTG+56YY5Eyj4BImlW5mIGV8xAbgW1QsBhPks5Un9MQ6fRol2j6FdOb+nshZbMw4Dm1nzHBoFmtT87+an2F0GeRCpRmC4vOPokxSTOj0ftoXGjjKsQXGtbC7Uj5kmnG0KVVsCN7iycvQPqt7lu/Oa43rIo4yOSLH5JR45II0yC1pkhbhJCHP5JW8Oei8OO/Ox7y15BQzh+SPnM8fG+OQeQ==</latexit>

Page 133: COMP 414/514: Optimization - Algorithms, Complexity and

Matrices

- Matrix singular value decomposition:

r {m,n}<latexit sha1_base64="RyyT3zeYAEA4tcZj4lBVYOs9IVw=">AAAB+HicbZDLSgMxFIbPeK310lGXboJFcCFlRgRdFt24rGAv0BlKJs20oUlmTDJCHfokblwo4tZHcefbmLaz0NYfAh//OYdz8kcpZ9p43rezsrq2vrFZ2ipv7+zuVdz9g5ZOMkVokyQ8UZ0Ia8qZpE3DDKedVFEsIk7b0ehmWm8/UqVZIu/NOKWhwAPJYkawsVbPrSgUcPqAglycIRlMem7Vq3kzoWXwC6hCoUbP/Qr6CckElYZwrHXX91IT5lgZRjidlINM0xSTER7QrkWJBdVhPjt8gk6s00dxouyTBs3c3xM5FlqPRWQ7BTZDvVibmv/VupmJr8KcyTQzVJL5ojjjyCRomgLqM0WJ4WMLmChmb0VkiBUmxmZVtiH4i19ehtZ5zbd8d1GtXxdxlOAIjuEUfLiEOtxCA5pAIINneIU358l5cd6dj3nrilPMHMIfOZ8/oHqSaA==</latexit><latexit sha1_base64="RyyT3zeYAEA4tcZj4lBVYOs9IVw=">AAAB+HicbZDLSgMxFIbPeK310lGXboJFcCFlRgRdFt24rGAv0BlKJs20oUlmTDJCHfokblwo4tZHcefbmLaz0NYfAh//OYdz8kcpZ9p43rezsrq2vrFZ2ipv7+zuVdz9g5ZOMkVokyQ8UZ0Ia8qZpE3DDKedVFEsIk7b0ehmWm8/UqVZIu/NOKWhwAPJYkawsVbPrSgUcPqAglycIRlMem7Vq3kzoWXwC6hCoUbP/Qr6CckElYZwrHXX91IT5lgZRjidlINM0xSTER7QrkWJBdVhPjt8gk6s00dxouyTBs3c3xM5FlqPRWQ7BTZDvVibmv/VupmJr8KcyTQzVJL5ojjjyCRomgLqM0WJ4WMLmChmb0VkiBUmxmZVtiH4i19ehtZ5zbd8d1GtXxdxlOAIjuEUfLiEOtxCA5pAIINneIU358l5cd6dj3nrilPMHMIfOZ8/oHqSaA==</latexit><latexit sha1_base64="RyyT3zeYAEA4tcZj4lBVYOs9IVw=">AAAB+HicbZDLSgMxFIbPeK310lGXboJFcCFlRgRdFt24rGAv0BlKJs20oUlmTDJCHfokblwo4tZHcefbmLaz0NYfAh//OYdz8kcpZ9p43rezsrq2vrFZ2ipv7+zuVdz9g5ZOMkVokyQ8UZ0Ia8qZpE3DDKedVFEsIk7b0ehmWm8/UqVZIu/NOKWhwAPJYkawsVbPrSgUcPqAglycIRlMem7Vq3kzoWXwC6hCoUbP/Qr6CckElYZwrHXX91IT5lgZRjidlINM0xSTER7QrkWJBdVhPjt8gk6s00dxouyTBs3c3xM5FlqPRWQ7BTZDvVibmv/VupmJr8KcyTQzVJL5ojjjyCRomgLqM0WJ4WMLmChmb0VkiBUmxmZVtiH4i19ehtZ5zbd8d1GtXxdxlOAIjuEUfLiEOtxCA5pAIINneIU358l5cd6dj3nrilPMHMIfOZ8/oHqSaA==</latexit><latexit sha1_base64="RyyT3zeYAEA4tcZj4lBVYOs9IVw=">AAAB+HicbZDLSgMxFIbPeK310lGXboJFcCFlRgRdFt24rGAv0BlKJs20oUlmTDJCHfokblwo4tZHcefbmLaz0NYfAh//OYdz8kcpZ9p43rezsrq2vrFZ2ipv7+zuVdz9g5ZOMkVokyQ8UZ0Ia8qZpE3DDKedVFEsIk7b0ehmWm8/UqVZIu/NOKWhwAPJYkawsVbPrSgUcPqAglycIRlMem7Vq3kzoWXwC6hCoUbP/Qr6CckElYZwrHXX91IT5lgZRjidlINM0xSTER7QrkWJBdVhPjt8gk6s00dxouyTBs3c3xM5FlqPRWQ7BTZDvVibmv/VupmJr8KcyTQzVJL5ojjjyCRomgLqM0WJ4WMLmChmb0VkiBUmxmZVtiH4i19ehtZ5zbd8d1GtXxdxlOAIjuEUfLiEOtxCA5pAIINneIU358l5cd6dj3nrilPMHMIfOZ8/oHqSaA==</latexit>

A 2 Rm⇥n<latexit sha1_base64="MxwCSL/gyMr0pbYAXqcODXgj2Pw=">AAACBnicbVBNS8NAEJ34WetX1KMIi0XwVBIR9Fj14rGK/YAmls120y7dbMLuRighJy/+FS8eFPHqb/Dmv3HT9qCtDwYe780wMy9IOFPacb6thcWl5ZXV0lp5fWNza9ve2W2qOJWENkjMY9kOsKKcCdrQTHPaTiTFUcBpKxheFX7rgUrFYnGnRwn1I9wXLGQEayN17YML5DGBvAjrQRBkt/l9FiFPs4gqJPKuXXGqzhhonrhTUoEp6l37y+vFJI2o0IRjpTquk2g/w1Izwmle9lJFE0yGuE87hgps9vjZ+I0cHRmlh8JYmhIajdXfExmOlBpFgekszlWzXiH+53VSHZ77GRNJqqkgk0VhypGOUZEJ6jFJieYjQzCRzNyKyABLTLRJrmxCcGdfnifNk6pr+M1ppXY5jaME+3AIx+DCGdTgGurQAAKP8Ayv8GY9WS/Wu/UxaV2wpjN78AfW5w+D2JiE</latexit><latexit sha1_base64="MxwCSL/gyMr0pbYAXqcODXgj2Pw=">AAACBnicbVBNS8NAEJ34WetX1KMIi0XwVBIR9Fj14rGK/YAmls120y7dbMLuRighJy/+FS8eFPHqb/Dmv3HT9qCtDwYe780wMy9IOFPacb6thcWl5ZXV0lp5fWNza9ve2W2qOJWENkjMY9kOsKKcCdrQTHPaTiTFUcBpKxheFX7rgUrFYnGnRwn1I9wXLGQEayN17YML5DGBvAjrQRBkt/l9FiFPs4gqJPKuXXGqzhhonrhTUoEp6l37y+vFJI2o0IRjpTquk2g/w1Izwmle9lJFE0yGuE87hgps9vjZ+I0cHRmlh8JYmhIajdXfExmOlBpFgekszlWzXiH+53VSHZ77GRNJqqkgk0VhypGOUZEJ6jFJieYjQzCRzNyKyABLTLRJrmxCcGdfnifNk6pr+M1ppXY5jaME+3AIx+DCGdTgGurQAAKP8Ayv8GY9WS/Wu/UxaV2wpjN78AfW5w+D2JiE</latexit><latexit sha1_base64="MxwCSL/gyMr0pbYAXqcODXgj2Pw=">AAACBnicbVBNS8NAEJ34WetX1KMIi0XwVBIR9Fj14rGK/YAmls120y7dbMLuRighJy/+FS8eFPHqb/Dmv3HT9qCtDwYe780wMy9IOFPacb6thcWl5ZXV0lp5fWNza9ve2W2qOJWENkjMY9kOsKKcCdrQTHPaTiTFUcBpKxheFX7rgUrFYnGnRwn1I9wXLGQEayN17YML5DGBvAjrQRBkt/l9FiFPs4gqJPKuXXGqzhhonrhTUoEp6l37y+vFJI2o0IRjpTquk2g/w1Izwmle9lJFE0yGuE87hgps9vjZ+I0cHRmlh8JYmhIajdXfExmOlBpFgekszlWzXiH+53VSHZ77GRNJqqkgk0VhypGOUZEJ6jFJieYjQzCRzNyKyABLTLRJrmxCcGdfnifNk6pr+M1ppXY5jaME+3AIx+DCGdTgGurQAAKP8Ayv8GY9WS/Wu/UxaV2wpjN78AfW5w+D2JiE</latexit><latexit sha1_base64="MxwCSL/gyMr0pbYAXqcODXgj2Pw=">AAACBnicbVBNS8NAEJ34WetX1KMIi0XwVBIR9Fj14rGK/YAmls120y7dbMLuRighJy/+FS8eFPHqb/Dmv3HT9qCtDwYe780wMy9IOFPacb6thcWl5ZXV0lp5fWNza9ve2W2qOJWENkjMY9kOsKKcCdrQTHPaTiTFUcBpKxheFX7rgUrFYnGnRwn1I9wXLGQEayN17YML5DGBvAjrQRBkt/l9FiFPs4gqJPKuXXGqzhhonrhTUoEp6l37y+vFJI2o0IRjpTquk2g/w1Izwmle9lJFE0yGuE87hgps9vjZ+I0cHRmlh8JYmhIajdXfExmOlBpFgekszlWzXiH+53VSHZ77GRNJqqkgk0VhypGOUZEJ6jFJieYjQzCRzNyKyABLTLRJrmxCcGdfnifNk6pr+M1ppXY5jaME+3AIx+DCGdTgGurQAAKP8Ayv8GY9WS/Wu/UxaV2wpjN78AfW5w+D2JiE</latexit>

A = U⌃V > =rX

i=1

�iuiv>i , U 2 Rm⇥r,⌃ 2 Rr⇥r, V 2 Rn⇥r

<latexit sha1_base64="lKyZjiBFpx910sCGQFpkkzLMSmw=">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</latexit><latexit sha1_base64="lKyZjiBFpx910sCGQFpkkzLMSmw=">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</latexit><latexit sha1_base64="lKyZjiBFpx910sCGQFpkkzLMSmw=">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</latexit><latexit sha1_base64="lKyZjiBFpx910sCGQFpkkzLMSmw=">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</latexit>

Page 134: COMP 414/514: Optimization - Algorithms, Complexity and

Matrices

- Matrix singular value decomposition:

r {m,n}<latexit sha1_base64="RyyT3zeYAEA4tcZj4lBVYOs9IVw=">AAAB+HicbZDLSgMxFIbPeK310lGXboJFcCFlRgRdFt24rGAv0BlKJs20oUlmTDJCHfokblwo4tZHcefbmLaz0NYfAh//OYdz8kcpZ9p43rezsrq2vrFZ2ipv7+zuVdz9g5ZOMkVokyQ8UZ0Ia8qZpE3DDKedVFEsIk7b0ehmWm8/UqVZIu/NOKWhwAPJYkawsVbPrSgUcPqAglycIRlMem7Vq3kzoWXwC6hCoUbP/Qr6CckElYZwrHXX91IT5lgZRjidlINM0xSTER7QrkWJBdVhPjt8gk6s00dxouyTBs3c3xM5FlqPRWQ7BTZDvVibmv/VupmJr8KcyTQzVJL5ojjjyCRomgLqM0WJ4WMLmChmb0VkiBUmxmZVtiH4i19ehtZ5zbd8d1GtXxdxlOAIjuEUfLiEOtxCA5pAIINneIU358l5cd6dj3nrilPMHMIfOZ8/oHqSaA==</latexit><latexit sha1_base64="RyyT3zeYAEA4tcZj4lBVYOs9IVw=">AAAB+HicbZDLSgMxFIbPeK310lGXboJFcCFlRgRdFt24rGAv0BlKJs20oUlmTDJCHfokblwo4tZHcefbmLaz0NYfAh//OYdz8kcpZ9p43rezsrq2vrFZ2ipv7+zuVdz9g5ZOMkVokyQ8UZ0Ia8qZpE3DDKedVFEsIk7b0ehmWm8/UqVZIu/NOKWhwAPJYkawsVbPrSgUcPqAglycIRlMem7Vq3kzoWXwC6hCoUbP/Qr6CckElYZwrHXX91IT5lgZRjidlINM0xSTER7QrkWJBdVhPjt8gk6s00dxouyTBs3c3xM5FlqPRWQ7BTZDvVibmv/VupmJr8KcyTQzVJL5ojjjyCRomgLqM0WJ4WMLmChmb0VkiBUmxmZVtiH4i19ehtZ5zbd8d1GtXxdxlOAIjuEUfLiEOtxCA5pAIINneIU358l5cd6dj3nrilPMHMIfOZ8/oHqSaA==</latexit><latexit sha1_base64="RyyT3zeYAEA4tcZj4lBVYOs9IVw=">AAAB+HicbZDLSgMxFIbPeK310lGXboJFcCFlRgRdFt24rGAv0BlKJs20oUlmTDJCHfokblwo4tZHcefbmLaz0NYfAh//OYdz8kcpZ9p43rezsrq2vrFZ2ipv7+zuVdz9g5ZOMkVokyQ8UZ0Ia8qZpE3DDKedVFEsIk7b0ehmWm8/UqVZIu/NOKWhwAPJYkawsVbPrSgUcPqAglycIRlMem7Vq3kzoWXwC6hCoUbP/Qr6CckElYZwrHXX91IT5lgZRjidlINM0xSTER7QrkWJBdVhPjt8gk6s00dxouyTBs3c3xM5FlqPRWQ7BTZDvVibmv/VupmJr8KcyTQzVJL5ojjjyCRomgLqM0WJ4WMLmChmb0VkiBUmxmZVtiH4i19ehtZ5zbd8d1GtXxdxlOAIjuEUfLiEOtxCA5pAIINneIU358l5cd6dj3nrilPMHMIfOZ8/oHqSaA==</latexit><latexit sha1_base64="RyyT3zeYAEA4tcZj4lBVYOs9IVw=">AAAB+HicbZDLSgMxFIbPeK310lGXboJFcCFlRgRdFt24rGAv0BlKJs20oUlmTDJCHfokblwo4tZHcefbmLaz0NYfAh//OYdz8kcpZ9p43rezsrq2vrFZ2ipv7+zuVdz9g5ZOMkVokyQ8UZ0Ia8qZpE3DDKedVFEsIk7b0ehmWm8/UqVZIu/NOKWhwAPJYkawsVbPrSgUcPqAglycIRlMem7Vq3kzoWXwC6hCoUbP/Qr6CckElYZwrHXX91IT5lgZRjidlINM0xSTER7QrkWJBdVhPjt8gk6s00dxouyTBs3c3xM5FlqPRWQ7BTZDvVibmv/VupmJr8KcyTQzVJL5ojjjyCRomgLqM0WJ4WMLmChmb0VkiBUmxmZVtiH4i19ehtZ5zbd8d1GtXxdxlOAIjuEUfLiEOtxCA5pAIINneIU358l5cd6dj3nrilPMHMIfOZ8/oHqSaA==</latexit>

A 2 Rm⇥n<latexit sha1_base64="MxwCSL/gyMr0pbYAXqcODXgj2Pw=">AAACBnicbVBNS8NAEJ34WetX1KMIi0XwVBIR9Fj14rGK/YAmls120y7dbMLuRighJy/+FS8eFPHqb/Dmv3HT9qCtDwYe780wMy9IOFPacb6thcWl5ZXV0lp5fWNza9ve2W2qOJWENkjMY9kOsKKcCdrQTHPaTiTFUcBpKxheFX7rgUrFYnGnRwn1I9wXLGQEayN17YML5DGBvAjrQRBkt/l9FiFPs4gqJPKuXXGqzhhonrhTUoEp6l37y+vFJI2o0IRjpTquk2g/w1Izwmle9lJFE0yGuE87hgps9vjZ+I0cHRmlh8JYmhIajdXfExmOlBpFgekszlWzXiH+53VSHZ77GRNJqqkgk0VhypGOUZEJ6jFJieYjQzCRzNyKyABLTLRJrmxCcGdfnifNk6pr+M1ppXY5jaME+3AIx+DCGdTgGurQAAKP8Ayv8GY9WS/Wu/UxaV2wpjN78AfW5w+D2JiE</latexit><latexit sha1_base64="MxwCSL/gyMr0pbYAXqcODXgj2Pw=">AAACBnicbVBNS8NAEJ34WetX1KMIi0XwVBIR9Fj14rGK/YAmls120y7dbMLuRighJy/+FS8eFPHqb/Dmv3HT9qCtDwYe780wMy9IOFPacb6thcWl5ZXV0lp5fWNza9ve2W2qOJWENkjMY9kOsKKcCdrQTHPaTiTFUcBpKxheFX7rgUrFYnGnRwn1I9wXLGQEayN17YML5DGBvAjrQRBkt/l9FiFPs4gqJPKuXXGqzhhonrhTUoEp6l37y+vFJI2o0IRjpTquk2g/w1Izwmle9lJFE0yGuE87hgps9vjZ+I0cHRmlh8JYmhIajdXfExmOlBpFgekszlWzXiH+53VSHZ77GRNJqqkgk0VhypGOUZEJ6jFJieYjQzCRzNyKyABLTLRJrmxCcGdfnifNk6pr+M1ppXY5jaME+3AIx+DCGdTgGurQAAKP8Ayv8GY9WS/Wu/UxaV2wpjN78AfW5w+D2JiE</latexit><latexit sha1_base64="MxwCSL/gyMr0pbYAXqcODXgj2Pw=">AAACBnicbVBNS8NAEJ34WetX1KMIi0XwVBIR9Fj14rGK/YAmls120y7dbMLuRighJy/+FS8eFPHqb/Dmv3HT9qCtDwYe780wMy9IOFPacb6thcWl5ZXV0lp5fWNza9ve2W2qOJWENkjMY9kOsKKcCdrQTHPaTiTFUcBpKxheFX7rgUrFYnGnRwn1I9wXLGQEayN17YML5DGBvAjrQRBkt/l9FiFPs4gqJPKuXXGqzhhonrhTUoEp6l37y+vFJI2o0IRjpTquk2g/w1Izwmle9lJFE0yGuE87hgps9vjZ+I0cHRmlh8JYmhIajdXfExmOlBpFgekszlWzXiH+53VSHZ77GRNJqqkgk0VhypGOUZEJ6jFJieYjQzCRzNyKyABLTLRJrmxCcGdfnifNk6pr+M1ppXY5jaME+3AIx+DCGdTgGurQAAKP8Ayv8GY9WS/Wu/UxaV2wpjN78AfW5w+D2JiE</latexit><latexit sha1_base64="MxwCSL/gyMr0pbYAXqcODXgj2Pw=">AAACBnicbVBNS8NAEJ34WetX1KMIi0XwVBIR9Fj14rGK/YAmls120y7dbMLuRighJy/+FS8eFPHqb/Dmv3HT9qCtDwYe780wMy9IOFPacb6thcWl5ZXV0lp5fWNza9ve2W2qOJWENkjMY9kOsKKcCdrQTHPaTiTFUcBpKxheFX7rgUrFYnGnRwn1I9wXLGQEayN17YML5DGBvAjrQRBkt/l9FiFPs4gqJPKuXXGqzhhonrhTUoEp6l37y+vFJI2o0IRjpTquk2g/w1Izwmle9lJFE0yGuE87hgps9vjZ+I0cHRmlh8JYmhIajdXfExmOlBpFgekszlWzXiH+53VSHZ77GRNJqqkgk0VhypGOUZEJ6jFJieYjQzCRzNyKyABLTLRJrmxCcGdfnifNk6pr+M1ppXY5jaME+3AIx+DCGdTgGurQAAKP8Ayv8GY9WS/Wu/UxaV2wpjN78AfW5w+D2JiE</latexit>

A = U⌃V > =rX

i=1

�iuiv>i , U 2 Rm⇥r,⌃ 2 Rr⇥r, V 2 Rn⇥r

<latexit sha1_base64="lKyZjiBFpx910sCGQFpkkzLMSmw=">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</latexit><latexit sha1_base64="lKyZjiBFpx910sCGQFpkkzLMSmw=">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</latexit><latexit sha1_base64="lKyZjiBFpx910sCGQFpkkzLMSmw=">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</latexit><latexit sha1_base64="lKyZjiBFpx910sCGQFpkkzLMSmw=">AAAChXicbVFNT9tAEF27fKThK9BjLyMiJA6AbIRoL4iUXnqkgANSnFjrzSas2F1bu2OkyHJ/SX8Vt/6broOl0oSRRnr73pud2dk0l8JiEPzx/A8rq2vrrY/tjc2t7Z3O7l7fZoVhPGKZzMxDSi2XQvMIBUr+kBtOVSr5ffr0vdbvn7mxItN3OMv5UNGpFhPBKDoq6fz+BhcQxbdiqij0RzFmuSNiW6ikFA6F1ci4Yy0nAgqXz4mY247gVwSx0BArio9pWt5Uo1JBjEJxC6Y6gubWRY956+kvyfqfnHS6wUkwD1gGYQO6pInrpPMSjzNWKK6RSWrtIAxyHJbUoGCSV+24sDyn7IlO+cBBTV2fYTnfYgUHjhnDJDMuNcKcfVtRUmXtTKXOWY9rF7WafE8bFDj5OiyFzgvkmr02mhQSMIP6S2AsDGcoZw5QZoSbFdgjNZSh+7i2W0K4+ORl0D89CR3+edbtXTXraJHPZJ8ckpB8IT3yg1yTiDDP9w690Dv11/1j/8w/f7X6XlPzifwX/uVfKiHBBg==</latexit>

- rank(A) = r min{m,n}<latexit sha1_base64="L/IAKAyqFfjh9StE51AH7q6v9V4=">AAACDXicbZBNSwMxEIazflu/qh69BKtQQWRXBL0IVS8eFWwVuqVk02kbmmTXZFYsS/+AF/+KFw+KePXuzX9j+nHQ6guBh3dmmMwbJVJY9P0vb2Jyanpmdm4+t7C4tLySX12r2Dg1HMo8lrG5iZgFKTSUUaCEm8QAU5GE66hz1q9f34GxItZX2E2gplhLi6bgDJ1Vz2+FCPeYGaY7veLJDj2mhoYSbmmohA4ztUt12KvnC/6ePxD9C8EICmSki3r+M2zEPFWgkUtmbTXwE6xlzKDgEnq5MLWQMN5hLag61EyBrWWDa3p02zkN2oyNexrpwP05kTFlbVdFrlMxbNvxWt/8r1ZNsXlUy4ROUgTNh4uaqaQY0340tCEMcJRdB4wb4f5KeZsZxtEFmHMhBOMn/4XK/l7g+PKgUDodxTFHNsgmKZKAHJISOScXpEw4eSBP5IW8eo/es/fmvQ9bJ7zRzDr5Je/jG7h8mq4=</latexit><latexit sha1_base64="L/IAKAyqFfjh9StE51AH7q6v9V4=">AAACDXicbZBNSwMxEIazflu/qh69BKtQQWRXBL0IVS8eFWwVuqVk02kbmmTXZFYsS/+AF/+KFw+KePXuzX9j+nHQ6guBh3dmmMwbJVJY9P0vb2Jyanpmdm4+t7C4tLySX12r2Dg1HMo8lrG5iZgFKTSUUaCEm8QAU5GE66hz1q9f34GxItZX2E2gplhLi6bgDJ1Vz2+FCPeYGaY7veLJDj2mhoYSbmmohA4ztUt12KvnC/6ePxD9C8EICmSki3r+M2zEPFWgkUtmbTXwE6xlzKDgEnq5MLWQMN5hLag61EyBrWWDa3p02zkN2oyNexrpwP05kTFlbVdFrlMxbNvxWt/8r1ZNsXlUy4ROUgTNh4uaqaQY0340tCEMcJRdB4wb4f5KeZsZxtEFmHMhBOMn/4XK/l7g+PKgUDodxTFHNsgmKZKAHJISOScXpEw4eSBP5IW8eo/es/fmvQ9bJ7zRzDr5Je/jG7h8mq4=</latexit><latexit sha1_base64="L/IAKAyqFfjh9StE51AH7q6v9V4=">AAACDXicbZBNSwMxEIazflu/qh69BKtQQWRXBL0IVS8eFWwVuqVk02kbmmTXZFYsS/+AF/+KFw+KePXuzX9j+nHQ6guBh3dmmMwbJVJY9P0vb2Jyanpmdm4+t7C4tLySX12r2Dg1HMo8lrG5iZgFKTSUUaCEm8QAU5GE66hz1q9f34GxItZX2E2gplhLi6bgDJ1Vz2+FCPeYGaY7veLJDj2mhoYSbmmohA4ztUt12KvnC/6ePxD9C8EICmSki3r+M2zEPFWgkUtmbTXwE6xlzKDgEnq5MLWQMN5hLag61EyBrWWDa3p02zkN2oyNexrpwP05kTFlbVdFrlMxbNvxWt/8r1ZNsXlUy4ROUgTNh4uaqaQY0340tCEMcJRdB4wb4f5KeZsZxtEFmHMhBOMn/4XK/l7g+PKgUDodxTFHNsgmKZKAHJISOScXpEw4eSBP5IW8eo/es/fmvQ9bJ7zRzDr5Je/jG7h8mq4=</latexit><latexit sha1_base64="L/IAKAyqFfjh9StE51AH7q6v9V4=">AAACDXicbZBNSwMxEIazflu/qh69BKtQQWRXBL0IVS8eFWwVuqVk02kbmmTXZFYsS/+AF/+KFw+KePXuzX9j+nHQ6guBh3dmmMwbJVJY9P0vb2Jyanpmdm4+t7C4tLySX12r2Dg1HMo8lrG5iZgFKTSUUaCEm8QAU5GE66hz1q9f34GxItZX2E2gplhLi6bgDJ1Vz2+FCPeYGaY7veLJDj2mhoYSbmmohA4ztUt12KvnC/6ePxD9C8EICmSki3r+M2zEPFWgkUtmbTXwE6xlzKDgEnq5MLWQMN5hLag61EyBrWWDa3p02zkN2oyNexrpwP05kTFlbVdFrlMxbNvxWt/8r1ZNsXlUy4ROUgTNh4uaqaQY0340tCEMcJRdB4wb4f5KeZsZxtEFmHMhBOMn/4XK/l7g+PKgUDodxTFHNsgmKZKAHJISOScXpEw4eSBP5IW8eo/es/fmvQ9bJ7zRzDr5Je/jG7h8mq4=</latexit>

Page 135: COMP 414/514: Optimization - Algorithms, Complexity and

Matrices

- Matrix singular value decomposition:

r {m,n}<latexit sha1_base64="RyyT3zeYAEA4tcZj4lBVYOs9IVw=">AAAB+HicbZDLSgMxFIbPeK310lGXboJFcCFlRgRdFt24rGAv0BlKJs20oUlmTDJCHfokblwo4tZHcefbmLaz0NYfAh//OYdz8kcpZ9p43rezsrq2vrFZ2ipv7+zuVdz9g5ZOMkVokyQ8UZ0Ia8qZpE3DDKedVFEsIk7b0ehmWm8/UqVZIu/NOKWhwAPJYkawsVbPrSgUcPqAglycIRlMem7Vq3kzoWXwC6hCoUbP/Qr6CckElYZwrHXX91IT5lgZRjidlINM0xSTER7QrkWJBdVhPjt8gk6s00dxouyTBs3c3xM5FlqPRWQ7BTZDvVibmv/VupmJr8KcyTQzVJL5ojjjyCRomgLqM0WJ4WMLmChmb0VkiBUmxmZVtiH4i19ehtZ5zbd8d1GtXxdxlOAIjuEUfLiEOtxCA5pAIINneIU358l5cd6dj3nrilPMHMIfOZ8/oHqSaA==</latexit><latexit sha1_base64="RyyT3zeYAEA4tcZj4lBVYOs9IVw=">AAAB+HicbZDLSgMxFIbPeK310lGXboJFcCFlRgRdFt24rGAv0BlKJs20oUlmTDJCHfokblwo4tZHcefbmLaz0NYfAh//OYdz8kcpZ9p43rezsrq2vrFZ2ipv7+zuVdz9g5ZOMkVokyQ8UZ0Ia8qZpE3DDKedVFEsIk7b0ehmWm8/UqVZIu/NOKWhwAPJYkawsVbPrSgUcPqAglycIRlMem7Vq3kzoWXwC6hCoUbP/Qr6CckElYZwrHXX91IT5lgZRjidlINM0xSTER7QrkWJBdVhPjt8gk6s00dxouyTBs3c3xM5FlqPRWQ7BTZDvVibmv/VupmJr8KcyTQzVJL5ojjjyCRomgLqM0WJ4WMLmChmb0VkiBUmxmZVtiH4i19ehtZ5zbd8d1GtXxdxlOAIjuEUfLiEOtxCA5pAIINneIU358l5cd6dj3nrilPMHMIfOZ8/oHqSaA==</latexit><latexit sha1_base64="RyyT3zeYAEA4tcZj4lBVYOs9IVw=">AAAB+HicbZDLSgMxFIbPeK310lGXboJFcCFlRgRdFt24rGAv0BlKJs20oUlmTDJCHfokblwo4tZHcefbmLaz0NYfAh//OYdz8kcpZ9p43rezsrq2vrFZ2ipv7+zuVdz9g5ZOMkVokyQ8UZ0Ia8qZpE3DDKedVFEsIk7b0ehmWm8/UqVZIu/NOKWhwAPJYkawsVbPrSgUcPqAglycIRlMem7Vq3kzoWXwC6hCoUbP/Qr6CckElYZwrHXX91IT5lgZRjidlINM0xSTER7QrkWJBdVhPjt8gk6s00dxouyTBs3c3xM5FlqPRWQ7BTZDvVibmv/VupmJr8KcyTQzVJL5ojjjyCRomgLqM0WJ4WMLmChmb0VkiBUmxmZVtiH4i19ehtZ5zbd8d1GtXxdxlOAIjuEUfLiEOtxCA5pAIINneIU358l5cd6dj3nrilPMHMIfOZ8/oHqSaA==</latexit><latexit sha1_base64="RyyT3zeYAEA4tcZj4lBVYOs9IVw=">AAAB+HicbZDLSgMxFIbPeK310lGXboJFcCFlRgRdFt24rGAv0BlKJs20oUlmTDJCHfokblwo4tZHcefbmLaz0NYfAh//OYdz8kcpZ9p43rezsrq2vrFZ2ipv7+zuVdz9g5ZOMkVokyQ8UZ0Ia8qZpE3DDKedVFEsIk7b0ehmWm8/UqVZIu/NOKWhwAPJYkawsVbPrSgUcPqAglycIRlMem7Vq3kzoWXwC6hCoUbP/Qr6CckElYZwrHXX91IT5lgZRjidlINM0xSTER7QrkWJBdVhPjt8gk6s00dxouyTBs3c3xM5FlqPRWQ7BTZDvVibmv/VupmJr8KcyTQzVJL5ojjjyCRomgLqM0WJ4WMLmChmb0VkiBUmxmZVtiH4i19ehtZ5zbd8d1GtXxdxlOAIjuEUfLiEOtxCA5pAIINneIU358l5cd6dj3nrilPMHMIfOZ8/oHqSaA==</latexit>

A 2 Rm⇥n<latexit sha1_base64="MxwCSL/gyMr0pbYAXqcODXgj2Pw=">AAACBnicbVBNS8NAEJ34WetX1KMIi0XwVBIR9Fj14rGK/YAmls120y7dbMLuRighJy/+FS8eFPHqb/Dmv3HT9qCtDwYe780wMy9IOFPacb6thcWl5ZXV0lp5fWNza9ve2W2qOJWENkjMY9kOsKKcCdrQTHPaTiTFUcBpKxheFX7rgUrFYnGnRwn1I9wXLGQEayN17YML5DGBvAjrQRBkt/l9FiFPs4gqJPKuXXGqzhhonrhTUoEp6l37y+vFJI2o0IRjpTquk2g/w1Izwmle9lJFE0yGuE87hgps9vjZ+I0cHRmlh8JYmhIajdXfExmOlBpFgekszlWzXiH+53VSHZ77GRNJqqkgk0VhypGOUZEJ6jFJieYjQzCRzNyKyABLTLRJrmxCcGdfnifNk6pr+M1ppXY5jaME+3AIx+DCGdTgGurQAAKP8Ayv8GY9WS/Wu/UxaV2wpjN78AfW5w+D2JiE</latexit><latexit sha1_base64="MxwCSL/gyMr0pbYAXqcODXgj2Pw=">AAACBnicbVBNS8NAEJ34WetX1KMIi0XwVBIR9Fj14rGK/YAmls120y7dbMLuRighJy/+FS8eFPHqb/Dmv3HT9qCtDwYe780wMy9IOFPacb6thcWl5ZXV0lp5fWNza9ve2W2qOJWENkjMY9kOsKKcCdrQTHPaTiTFUcBpKxheFX7rgUrFYnGnRwn1I9wXLGQEayN17YML5DGBvAjrQRBkt/l9FiFPs4gqJPKuXXGqzhhonrhTUoEp6l37y+vFJI2o0IRjpTquk2g/w1Izwmle9lJFE0yGuE87hgps9vjZ+I0cHRmlh8JYmhIajdXfExmOlBpFgekszlWzXiH+53VSHZ77GRNJqqkgk0VhypGOUZEJ6jFJieYjQzCRzNyKyABLTLRJrmxCcGdfnifNk6pr+M1ppXY5jaME+3AIx+DCGdTgGurQAAKP8Ayv8GY9WS/Wu/UxaV2wpjN78AfW5w+D2JiE</latexit><latexit sha1_base64="MxwCSL/gyMr0pbYAXqcODXgj2Pw=">AAACBnicbVBNS8NAEJ34WetX1KMIi0XwVBIR9Fj14rGK/YAmls120y7dbMLuRighJy/+FS8eFPHqb/Dmv3HT9qCtDwYe780wMy9IOFPacb6thcWl5ZXV0lp5fWNza9ve2W2qOJWENkjMY9kOsKKcCdrQTHPaTiTFUcBpKxheFX7rgUrFYnGnRwn1I9wXLGQEayN17YML5DGBvAjrQRBkt/l9FiFPs4gqJPKuXXGqzhhonrhTUoEp6l37y+vFJI2o0IRjpTquk2g/w1Izwmle9lJFE0yGuE87hgps9vjZ+I0cHRmlh8JYmhIajdXfExmOlBpFgekszlWzXiH+53VSHZ77GRNJqqkgk0VhypGOUZEJ6jFJieYjQzCRzNyKyABLTLRJrmxCcGdfnifNk6pr+M1ppXY5jaME+3AIx+DCGdTgGurQAAKP8Ayv8GY9WS/Wu/UxaV2wpjN78AfW5w+D2JiE</latexit><latexit sha1_base64="MxwCSL/gyMr0pbYAXqcODXgj2Pw=">AAACBnicbVBNS8NAEJ34WetX1KMIi0XwVBIR9Fj14rGK/YAmls120y7dbMLuRighJy/+FS8eFPHqb/Dmv3HT9qCtDwYe780wMy9IOFPacb6thcWl5ZXV0lp5fWNza9ve2W2qOJWENkjMY9kOsKKcCdrQTHPaTiTFUcBpKxheFX7rgUrFYnGnRwn1I9wXLGQEayN17YML5DGBvAjrQRBkt/l9FiFPs4gqJPKuXXGqzhhonrhTUoEp6l37y+vFJI2o0IRjpTquk2g/w1Izwmle9lJFE0yGuE87hgps9vjZ+I0cHRmlh8JYmhIajdXfExmOlBpFgekszlWzXiH+53VSHZ77GRNJqqkgk0VhypGOUZEJ6jFJieYjQzCRzNyKyABLTLRJrmxCcGdfnifNk6pr+M1ppXY5jaME+3AIx+DCGdTgGurQAAKP8Ayv8GY9WS/Wu/UxaV2wpjN78AfW5w+D2JiE</latexit>

A = U⌃V > =rX

i=1

�iuiv>i , U 2 Rm⇥r,⌃ 2 Rr⇥r, V 2 Rn⇥r

<latexit sha1_base64="lKyZjiBFpx910sCGQFpkkzLMSmw=">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</latexit><latexit sha1_base64="lKyZjiBFpx910sCGQFpkkzLMSmw=">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</latexit><latexit sha1_base64="lKyZjiBFpx910sCGQFpkkzLMSmw=">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</latexit><latexit sha1_base64="lKyZjiBFpx910sCGQFpkkzLMSmw=">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</latexit>

- rank(A) = r min{m,n}<latexit sha1_base64="L/IAKAyqFfjh9StE51AH7q6v9V4=">AAACDXicbZBNSwMxEIazflu/qh69BKtQQWRXBL0IVS8eFWwVuqVk02kbmmTXZFYsS/+AF/+KFw+KePXuzX9j+nHQ6guBh3dmmMwbJVJY9P0vb2Jyanpmdm4+t7C4tLySX12r2Dg1HMo8lrG5iZgFKTSUUaCEm8QAU5GE66hz1q9f34GxItZX2E2gplhLi6bgDJ1Vz2+FCPeYGaY7veLJDj2mhoYSbmmohA4ztUt12KvnC/6ePxD9C8EICmSki3r+M2zEPFWgkUtmbTXwE6xlzKDgEnq5MLWQMN5hLag61EyBrWWDa3p02zkN2oyNexrpwP05kTFlbVdFrlMxbNvxWt/8r1ZNsXlUy4ROUgTNh4uaqaQY0340tCEMcJRdB4wb4f5KeZsZxtEFmHMhBOMn/4XK/l7g+PKgUDodxTFHNsgmKZKAHJISOScXpEw4eSBP5IW8eo/es/fmvQ9bJ7zRzDr5Je/jG7h8mq4=</latexit><latexit sha1_base64="L/IAKAyqFfjh9StE51AH7q6v9V4=">AAACDXicbZBNSwMxEIazflu/qh69BKtQQWRXBL0IVS8eFWwVuqVk02kbmmTXZFYsS/+AF/+KFw+KePXuzX9j+nHQ6guBh3dmmMwbJVJY9P0vb2Jyanpmdm4+t7C4tLySX12r2Dg1HMo8lrG5iZgFKTSUUaCEm8QAU5GE66hz1q9f34GxItZX2E2gplhLi6bgDJ1Vz2+FCPeYGaY7veLJDj2mhoYSbmmohA4ztUt12KvnC/6ePxD9C8EICmSki3r+M2zEPFWgkUtmbTXwE6xlzKDgEnq5MLWQMN5hLag61EyBrWWDa3p02zkN2oyNexrpwP05kTFlbVdFrlMxbNvxWt/8r1ZNsXlUy4ROUgTNh4uaqaQY0340tCEMcJRdB4wb4f5KeZsZxtEFmHMhBOMn/4XK/l7g+PKgUDodxTFHNsgmKZKAHJISOScXpEw4eSBP5IW8eo/es/fmvQ9bJ7zRzDr5Je/jG7h8mq4=</latexit><latexit sha1_base64="L/IAKAyqFfjh9StE51AH7q6v9V4=">AAACDXicbZBNSwMxEIazflu/qh69BKtQQWRXBL0IVS8eFWwVuqVk02kbmmTXZFYsS/+AF/+KFw+KePXuzX9j+nHQ6guBh3dmmMwbJVJY9P0vb2Jyanpmdm4+t7C4tLySX12r2Dg1HMo8lrG5iZgFKTSUUaCEm8QAU5GE66hz1q9f34GxItZX2E2gplhLi6bgDJ1Vz2+FCPeYGaY7veLJDj2mhoYSbmmohA4ztUt12KvnC/6ePxD9C8EICmSki3r+M2zEPFWgkUtmbTXwE6xlzKDgEnq5MLWQMN5hLag61EyBrWWDa3p02zkN2oyNexrpwP05kTFlbVdFrlMxbNvxWt/8r1ZNsXlUy4ROUgTNh4uaqaQY0340tCEMcJRdB4wb4f5KeZsZxtEFmHMhBOMn/4XK/l7g+PKgUDodxTFHNsgmKZKAHJISOScXpEw4eSBP5IW8eo/es/fmvQ9bJ7zRzDr5Je/jG7h8mq4=</latexit><latexit sha1_base64="L/IAKAyqFfjh9StE51AH7q6v9V4=">AAACDXicbZBNSwMxEIazflu/qh69BKtQQWRXBL0IVS8eFWwVuqVk02kbmmTXZFYsS/+AF/+KFw+KePXuzX9j+nHQ6guBh3dmmMwbJVJY9P0vb2Jyanpmdm4+t7C4tLySX12r2Dg1HMo8lrG5iZgFKTSUUaCEm8QAU5GE66hz1q9f34GxItZX2E2gplhLi6bgDJ1Vz2+FCPeYGaY7veLJDj2mhoYSbmmohA4ztUt12KvnC/6ePxD9C8EICmSki3r+M2zEPFWgkUtmbTXwE6xlzKDgEnq5MLWQMN5hLag61EyBrWWDa3p02zkN2oyNexrpwP05kTFlbVdFrlMxbNvxWt/8r1ZNsXlUy4ROUgTNh4uaqaQY0340tCEMcJRdB4wb4f5KeZsZxtEFmHMhBOMn/4XK/l7g+PKgUDodxTFHNsgmKZKAHJISOScXpEw4eSBP5IW8eo/es/fmvQ9bJ7zRzDr5Je/jG7h8mq4=</latexit>

- are the left and right singular vectorsui 2 Rm, vi 2 Rn<latexit sha1_base64="KAAe+IfS96EJriixWDJslYBAwtI=">AAACF3icbZDLSsNAFIYn9VbrLerSzWARXEhJRNBl0Y3LKvYCbQyT6aQdOjMJM5NCCXkLN76KGxeKuNWdb+OkzULb/jDw8Z9zmHP+IGZUacf5sUorq2vrG+XNytb2zu6evX/QUlEiMWniiEWyEyBFGBWkqalmpBNLgnjASDsY3eT19phIRSPxoCcx8TgaCBpSjLSxfLuW+BT2qIA9jvQwCNL77DHl2RkcL/FF5ttVp+ZMBRfBLaAKCjV8+7vXj3DCidCYIaW6rhNrL0VSU8xIVuklisQIj9CAdA0KxIny0uldGTwxTh+GkTRPaDh1/06kiCs14YHpzLdU87XcXFbrJjq88lIq4kQTgWcfhQmDOoJ5SLBPJcGaTQwgLKnZFeIhkghrE2XFhODOn7wIrfOaa/juolq/LuIogyNwDE6BCy5BHdyCBmgCDJ7AC3gD79az9Wp9WJ+z1pJVzByCf7K+fgH9+Z/T</latexit><latexit sha1_base64="KAAe+IfS96EJriixWDJslYBAwtI=">AAACF3icbZDLSsNAFIYn9VbrLerSzWARXEhJRNBl0Y3LKvYCbQyT6aQdOjMJM5NCCXkLN76KGxeKuNWdb+OkzULb/jDw8Z9zmHP+IGZUacf5sUorq2vrG+XNytb2zu6evX/QUlEiMWniiEWyEyBFGBWkqalmpBNLgnjASDsY3eT19phIRSPxoCcx8TgaCBpSjLSxfLuW+BT2qIA9jvQwCNL77DHl2RkcL/FF5ttVp+ZMBRfBLaAKCjV8+7vXj3DCidCYIaW6rhNrL0VSU8xIVuklisQIj9CAdA0KxIny0uldGTwxTh+GkTRPaDh1/06kiCs14YHpzLdU87XcXFbrJjq88lIq4kQTgWcfhQmDOoJ5SLBPJcGaTQwgLKnZFeIhkghrE2XFhODOn7wIrfOaa/juolq/LuIogyNwDE6BCy5BHdyCBmgCDJ7AC3gD79az9Wp9WJ+z1pJVzByCf7K+fgH9+Z/T</latexit><latexit sha1_base64="KAAe+IfS96EJriixWDJslYBAwtI=">AAACF3icbZDLSsNAFIYn9VbrLerSzWARXEhJRNBl0Y3LKvYCbQyT6aQdOjMJM5NCCXkLN76KGxeKuNWdb+OkzULb/jDw8Z9zmHP+IGZUacf5sUorq2vrG+XNytb2zu6evX/QUlEiMWniiEWyEyBFGBWkqalmpBNLgnjASDsY3eT19phIRSPxoCcx8TgaCBpSjLSxfLuW+BT2qIA9jvQwCNL77DHl2RkcL/FF5ttVp+ZMBRfBLaAKCjV8+7vXj3DCidCYIaW6rhNrL0VSU8xIVuklisQIj9CAdA0KxIny0uldGTwxTh+GkTRPaDh1/06kiCs14YHpzLdU87XcXFbrJjq88lIq4kQTgWcfhQmDOoJ5SLBPJcGaTQwgLKnZFeIhkghrE2XFhODOn7wIrfOaa/juolq/LuIogyNwDE6BCy5BHdyCBmgCDJ7AC3gD79az9Wp9WJ+z1pJVzByCf7K+fgH9+Z/T</latexit><latexit sha1_base64="KAAe+IfS96EJriixWDJslYBAwtI=">AAACF3icbZDLSsNAFIYn9VbrLerSzWARXEhJRNBl0Y3LKvYCbQyT6aQdOjMJM5NCCXkLN76KGxeKuNWdb+OkzULb/jDw8Z9zmHP+IGZUacf5sUorq2vrG+XNytb2zu6evX/QUlEiMWniiEWyEyBFGBWkqalmpBNLgnjASDsY3eT19phIRSPxoCcx8TgaCBpSjLSxfLuW+BT2qIA9jvQwCNL77DHl2RkcL/FF5ttVp+ZMBRfBLaAKCjV8+7vXj3DCidCYIaW6rhNrL0VSU8xIVuklisQIj9CAdA0KxIny0uldGTwxTh+GkTRPaDh1/06kiCs14YHpzLdU87XcXFbrJjq88lIq4kQTgWcfhQmDOoJ5SLBPJcGaTQwgLKnZFeIhkghrE2XFhODOn7wIrfOaa/juolq/LuIogyNwDE6BCy5BHdyCBmgCDJ7AC3gD79az9Wp9WJ+z1pJVzByCf7K+fgH9+Z/T</latexit>

Page 136: COMP 414/514: Optimization - Algorithms, Complexity and

Matrices

- Matrix singular value decomposition:

r {m,n}<latexit sha1_base64="RyyT3zeYAEA4tcZj4lBVYOs9IVw=">AAAB+HicbZDLSgMxFIbPeK310lGXboJFcCFlRgRdFt24rGAv0BlKJs20oUlmTDJCHfokblwo4tZHcefbmLaz0NYfAh//OYdz8kcpZ9p43rezsrq2vrFZ2ipv7+zuVdz9g5ZOMkVokyQ8UZ0Ia8qZpE3DDKedVFEsIk7b0ehmWm8/UqVZIu/NOKWhwAPJYkawsVbPrSgUcPqAglycIRlMem7Vq3kzoWXwC6hCoUbP/Qr6CckElYZwrHXX91IT5lgZRjidlINM0xSTER7QrkWJBdVhPjt8gk6s00dxouyTBs3c3xM5FlqPRWQ7BTZDvVibmv/VupmJr8KcyTQzVJL5ojjjyCRomgLqM0WJ4WMLmChmb0VkiBUmxmZVtiH4i19ehtZ5zbd8d1GtXxdxlOAIjuEUfLiEOtxCA5pAIINneIU358l5cd6dj3nrilPMHMIfOZ8/oHqSaA==</latexit><latexit sha1_base64="RyyT3zeYAEA4tcZj4lBVYOs9IVw=">AAAB+HicbZDLSgMxFIbPeK310lGXboJFcCFlRgRdFt24rGAv0BlKJs20oUlmTDJCHfokblwo4tZHcefbmLaz0NYfAh//OYdz8kcpZ9p43rezsrq2vrFZ2ipv7+zuVdz9g5ZOMkVokyQ8UZ0Ia8qZpE3DDKedVFEsIk7b0ehmWm8/UqVZIu/NOKWhwAPJYkawsVbPrSgUcPqAglycIRlMem7Vq3kzoWXwC6hCoUbP/Qr6CckElYZwrHXX91IT5lgZRjidlINM0xSTER7QrkWJBdVhPjt8gk6s00dxouyTBs3c3xM5FlqPRWQ7BTZDvVibmv/VupmJr8KcyTQzVJL5ojjjyCRomgLqM0WJ4WMLmChmb0VkiBUmxmZVtiH4i19ehtZ5zbd8d1GtXxdxlOAIjuEUfLiEOtxCA5pAIINneIU358l5cd6dj3nrilPMHMIfOZ8/oHqSaA==</latexit><latexit sha1_base64="RyyT3zeYAEA4tcZj4lBVYOs9IVw=">AAAB+HicbZDLSgMxFIbPeK310lGXboJFcCFlRgRdFt24rGAv0BlKJs20oUlmTDJCHfokblwo4tZHcefbmLaz0NYfAh//OYdz8kcpZ9p43rezsrq2vrFZ2ipv7+zuVdz9g5ZOMkVokyQ8UZ0Ia8qZpE3DDKedVFEsIk7b0ehmWm8/UqVZIu/NOKWhwAPJYkawsVbPrSgUcPqAglycIRlMem7Vq3kzoWXwC6hCoUbP/Qr6CckElYZwrHXX91IT5lgZRjidlINM0xSTER7QrkWJBdVhPjt8gk6s00dxouyTBs3c3xM5FlqPRWQ7BTZDvVibmv/VupmJr8KcyTQzVJL5ojjjyCRomgLqM0WJ4WMLmChmb0VkiBUmxmZVtiH4i19ehtZ5zbd8d1GtXxdxlOAIjuEUfLiEOtxCA5pAIINneIU358l5cd6dj3nrilPMHMIfOZ8/oHqSaA==</latexit><latexit sha1_base64="RyyT3zeYAEA4tcZj4lBVYOs9IVw=">AAAB+HicbZDLSgMxFIbPeK310lGXboJFcCFlRgRdFt24rGAv0BlKJs20oUlmTDJCHfokblwo4tZHcefbmLaz0NYfAh//OYdz8kcpZ9p43rezsrq2vrFZ2ipv7+zuVdz9g5ZOMkVokyQ8UZ0Ia8qZpE3DDKedVFEsIk7b0ehmWm8/UqVZIu/NOKWhwAPJYkawsVbPrSgUcPqAglycIRlMem7Vq3kzoWXwC6hCoUbP/Qr6CckElYZwrHXX91IT5lgZRjidlINM0xSTER7QrkWJBdVhPjt8gk6s00dxouyTBs3c3xM5FlqPRWQ7BTZDvVibmv/VupmJr8KcyTQzVJL5ojjjyCRomgLqM0WJ4WMLmChmb0VkiBUmxmZVtiH4i19ehtZ5zbd8d1GtXxdxlOAIjuEUfLiEOtxCA5pAIINneIU358l5cd6dj3nrilPMHMIfOZ8/oHqSaA==</latexit>

A 2 Rm⇥n<latexit sha1_base64="MxwCSL/gyMr0pbYAXqcODXgj2Pw=">AAACBnicbVBNS8NAEJ34WetX1KMIi0XwVBIR9Fj14rGK/YAmls120y7dbMLuRighJy/+FS8eFPHqb/Dmv3HT9qCtDwYe780wMy9IOFPacb6thcWl5ZXV0lp5fWNza9ve2W2qOJWENkjMY9kOsKKcCdrQTHPaTiTFUcBpKxheFX7rgUrFYnGnRwn1I9wXLGQEayN17YML5DGBvAjrQRBkt/l9FiFPs4gqJPKuXXGqzhhonrhTUoEp6l37y+vFJI2o0IRjpTquk2g/w1Izwmle9lJFE0yGuE87hgps9vjZ+I0cHRmlh8JYmhIajdXfExmOlBpFgekszlWzXiH+53VSHZ77GRNJqqkgk0VhypGOUZEJ6jFJieYjQzCRzNyKyABLTLRJrmxCcGdfnifNk6pr+M1ppXY5jaME+3AIx+DCGdTgGurQAAKP8Ayv8GY9WS/Wu/UxaV2wpjN78AfW5w+D2JiE</latexit><latexit sha1_base64="MxwCSL/gyMr0pbYAXqcODXgj2Pw=">AAACBnicbVBNS8NAEJ34WetX1KMIi0XwVBIR9Fj14rGK/YAmls120y7dbMLuRighJy/+FS8eFPHqb/Dmv3HT9qCtDwYe780wMy9IOFPacb6thcWl5ZXV0lp5fWNza9ve2W2qOJWENkjMY9kOsKKcCdrQTHPaTiTFUcBpKxheFX7rgUrFYnGnRwn1I9wXLGQEayN17YML5DGBvAjrQRBkt/l9FiFPs4gqJPKuXXGqzhhonrhTUoEp6l37y+vFJI2o0IRjpTquk2g/w1Izwmle9lJFE0yGuE87hgps9vjZ+I0cHRmlh8JYmhIajdXfExmOlBpFgekszlWzXiH+53VSHZ77GRNJqqkgk0VhypGOUZEJ6jFJieYjQzCRzNyKyABLTLRJrmxCcGdfnifNk6pr+M1ppXY5jaME+3AIx+DCGdTgGurQAAKP8Ayv8GY9WS/Wu/UxaV2wpjN78AfW5w+D2JiE</latexit><latexit sha1_base64="MxwCSL/gyMr0pbYAXqcODXgj2Pw=">AAACBnicbVBNS8NAEJ34WetX1KMIi0XwVBIR9Fj14rGK/YAmls120y7dbMLuRighJy/+FS8eFPHqb/Dmv3HT9qCtDwYe780wMy9IOFPacb6thcWl5ZXV0lp5fWNza9ve2W2qOJWENkjMY9kOsKKcCdrQTHPaTiTFUcBpKxheFX7rgUrFYnGnRwn1I9wXLGQEayN17YML5DGBvAjrQRBkt/l9FiFPs4gqJPKuXXGqzhhonrhTUoEp6l37y+vFJI2o0IRjpTquk2g/w1Izwmle9lJFE0yGuE87hgps9vjZ+I0cHRmlh8JYmhIajdXfExmOlBpFgekszlWzXiH+53VSHZ77GRNJqqkgk0VhypGOUZEJ6jFJieYjQzCRzNyKyABLTLRJrmxCcGdfnifNk6pr+M1ppXY5jaME+3AIx+DCGdTgGurQAAKP8Ayv8GY9WS/Wu/UxaV2wpjN78AfW5w+D2JiE</latexit><latexit sha1_base64="MxwCSL/gyMr0pbYAXqcODXgj2Pw=">AAACBnicbVBNS8NAEJ34WetX1KMIi0XwVBIR9Fj14rGK/YAmls120y7dbMLuRighJy/+FS8eFPHqb/Dmv3HT9qCtDwYe780wMy9IOFPacb6thcWl5ZXV0lp5fWNza9ve2W2qOJWENkjMY9kOsKKcCdrQTHPaTiTFUcBpKxheFX7rgUrFYnGnRwn1I9wXLGQEayN17YML5DGBvAjrQRBkt/l9FiFPs4gqJPKuXXGqzhhonrhTUoEp6l37y+vFJI2o0IRjpTquk2g/w1Izwmle9lJFE0yGuE87hgps9vjZ+I0cHRmlh8JYmhIajdXfExmOlBpFgekszlWzXiH+53VSHZ77GRNJqqkgk0VhypGOUZEJ6jFJieYjQzCRzNyKyABLTLRJrmxCcGdfnifNk6pr+M1ppXY5jaME+3AIx+DCGdTgGurQAAKP8Ayv8GY9WS/Wu/UxaV2wpjN78AfW5w+D2JiE</latexit>

A = U⌃V > =rX

i=1

�iuiv>i , U 2 Rm⇥r,⌃ 2 Rr⇥r, V 2 Rn⇥r

<latexit sha1_base64="lKyZjiBFpx910sCGQFpkkzLMSmw=">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</latexit><latexit sha1_base64="lKyZjiBFpx910sCGQFpkkzLMSmw=">AAAChXicbVFNT9tAEF27fKThK9BjLyMiJA6AbIRoL4iUXnqkgANSnFjrzSas2F1bu2OkyHJ/SX8Vt/6broOl0oSRRnr73pud2dk0l8JiEPzx/A8rq2vrrY/tjc2t7Z3O7l7fZoVhPGKZzMxDSi2XQvMIBUr+kBtOVSr5ffr0vdbvn7mxItN3OMv5UNGpFhPBKDoq6fz+BhcQxbdiqij0RzFmuSNiW6ikFA6F1ci4Yy0nAgqXz4mY247gVwSx0BArio9pWt5Uo1JBjEJxC6Y6gubWRY956+kvyfqfnHS6wUkwD1gGYQO6pInrpPMSjzNWKK6RSWrtIAxyHJbUoGCSV+24sDyn7IlO+cBBTV2fYTnfYgUHjhnDJDMuNcKcfVtRUmXtTKXOWY9rF7WafE8bFDj5OiyFzgvkmr02mhQSMIP6S2AsDGcoZw5QZoSbFdgjNZSh+7i2W0K4+ORl0D89CR3+edbtXTXraJHPZJ8ckpB8IT3yg1yTiDDP9w690Dv11/1j/8w/f7X6XlPzifwX/uVfKiHBBg==</latexit><latexit sha1_base64="lKyZjiBFpx910sCGQFpkkzLMSmw=">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</latexit><latexit sha1_base64="lKyZjiBFpx910sCGQFpkkzLMSmw=">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</latexit>

- rank(A) = r min{m,n}<latexit sha1_base64="L/IAKAyqFfjh9StE51AH7q6v9V4=">AAACDXicbZBNSwMxEIazflu/qh69BKtQQWRXBL0IVS8eFWwVuqVk02kbmmTXZFYsS/+AF/+KFw+KePXuzX9j+nHQ6guBh3dmmMwbJVJY9P0vb2Jyanpmdm4+t7C4tLySX12r2Dg1HMo8lrG5iZgFKTSUUaCEm8QAU5GE66hz1q9f34GxItZX2E2gplhLi6bgDJ1Vz2+FCPeYGaY7veLJDj2mhoYSbmmohA4ztUt12KvnC/6ePxD9C8EICmSki3r+M2zEPFWgkUtmbTXwE6xlzKDgEnq5MLWQMN5hLag61EyBrWWDa3p02zkN2oyNexrpwP05kTFlbVdFrlMxbNvxWt/8r1ZNsXlUy4ROUgTNh4uaqaQY0340tCEMcJRdB4wb4f5KeZsZxtEFmHMhBOMn/4XK/l7g+PKgUDodxTFHNsgmKZKAHJISOScXpEw4eSBP5IW8eo/es/fmvQ9bJ7zRzDr5Je/jG7h8mq4=</latexit><latexit sha1_base64="L/IAKAyqFfjh9StE51AH7q6v9V4=">AAACDXicbZBNSwMxEIazflu/qh69BKtQQWRXBL0IVS8eFWwVuqVk02kbmmTXZFYsS/+AF/+KFw+KePXuzX9j+nHQ6guBh3dmmMwbJVJY9P0vb2Jyanpmdm4+t7C4tLySX12r2Dg1HMo8lrG5iZgFKTSUUaCEm8QAU5GE66hz1q9f34GxItZX2E2gplhLi6bgDJ1Vz2+FCPeYGaY7veLJDj2mhoYSbmmohA4ztUt12KvnC/6ePxD9C8EICmSki3r+M2zEPFWgkUtmbTXwE6xlzKDgEnq5MLWQMN5hLag61EyBrWWDa3p02zkN2oyNexrpwP05kTFlbVdFrlMxbNvxWt/8r1ZNsXlUy4ROUgTNh4uaqaQY0340tCEMcJRdB4wb4f5KeZsZxtEFmHMhBOMn/4XK/l7g+PKgUDodxTFHNsgmKZKAHJISOScXpEw4eSBP5IW8eo/es/fmvQ9bJ7zRzDr5Je/jG7h8mq4=</latexit><latexit sha1_base64="L/IAKAyqFfjh9StE51AH7q6v9V4=">AAACDXicbZBNSwMxEIazflu/qh69BKtQQWRXBL0IVS8eFWwVuqVk02kbmmTXZFYsS/+AF/+KFw+KePXuzX9j+nHQ6guBh3dmmMwbJVJY9P0vb2Jyanpmdm4+t7C4tLySX12r2Dg1HMo8lrG5iZgFKTSUUaCEm8QAU5GE66hz1q9f34GxItZX2E2gplhLi6bgDJ1Vz2+FCPeYGaY7veLJDj2mhoYSbmmohA4ztUt12KvnC/6ePxD9C8EICmSki3r+M2zEPFWgkUtmbTXwE6xlzKDgEnq5MLWQMN5hLag61EyBrWWDa3p02zkN2oyNexrpwP05kTFlbVdFrlMxbNvxWt/8r1ZNsXlUy4ROUgTNh4uaqaQY0340tCEMcJRdB4wb4f5KeZsZxtEFmHMhBOMn/4XK/l7g+PKgUDodxTFHNsgmKZKAHJISOScXpEw4eSBP5IW8eo/es/fmvQ9bJ7zRzDr5Je/jG7h8mq4=</latexit><latexit sha1_base64="L/IAKAyqFfjh9StE51AH7q6v9V4=">AAACDXicbZBNSwMxEIazflu/qh69BKtQQWRXBL0IVS8eFWwVuqVk02kbmmTXZFYsS/+AF/+KFw+KePXuzX9j+nHQ6guBh3dmmMwbJVJY9P0vb2Jyanpmdm4+t7C4tLySX12r2Dg1HMo8lrG5iZgFKTSUUaCEm8QAU5GE66hz1q9f34GxItZX2E2gplhLi6bgDJ1Vz2+FCPeYGaY7veLJDj2mhoYSbmmohA4ztUt12KvnC/6ePxD9C8EICmSki3r+M2zEPFWgkUtmbTXwE6xlzKDgEnq5MLWQMN5hLag61EyBrWWDa3p02zkN2oyNexrpwP05kTFlbVdFrlMxbNvxWt/8r1ZNsXlUy4ROUgTNh4uaqaQY0340tCEMcJRdB4wb4f5KeZsZxtEFmHMhBOMn/4XK/l7g+PKgUDodxTFHNsgmKZKAHJISOScXpEw4eSBP5IW8eo/es/fmvQ9bJ7zRzDr5Je/jG7h8mq4=</latexit>

- are the left and right singular vectorsui 2 Rm, vi 2 Rn<latexit sha1_base64="KAAe+IfS96EJriixWDJslYBAwtI=">AAACF3icbZDLSsNAFIYn9VbrLerSzWARXEhJRNBl0Y3LKvYCbQyT6aQdOjMJM5NCCXkLN76KGxeKuNWdb+OkzULb/jDw8Z9zmHP+IGZUacf5sUorq2vrG+XNytb2zu6evX/QUlEiMWniiEWyEyBFGBWkqalmpBNLgnjASDsY3eT19phIRSPxoCcx8TgaCBpSjLSxfLuW+BT2qIA9jvQwCNL77DHl2RkcL/FF5ttVp+ZMBRfBLaAKCjV8+7vXj3DCidCYIaW6rhNrL0VSU8xIVuklisQIj9CAdA0KxIny0uldGTwxTh+GkTRPaDh1/06kiCs14YHpzLdU87XcXFbrJjq88lIq4kQTgWcfhQmDOoJ5SLBPJcGaTQwgLKnZFeIhkghrE2XFhODOn7wIrfOaa/juolq/LuIogyNwDE6BCy5BHdyCBmgCDJ7AC3gD79az9Wp9WJ+z1pJVzByCf7K+fgH9+Z/T</latexit><latexit sha1_base64="KAAe+IfS96EJriixWDJslYBAwtI=">AAACF3icbZDLSsNAFIYn9VbrLerSzWARXEhJRNBl0Y3LKvYCbQyT6aQdOjMJM5NCCXkLN76KGxeKuNWdb+OkzULb/jDw8Z9zmHP+IGZUacf5sUorq2vrG+XNytb2zu6evX/QUlEiMWniiEWyEyBFGBWkqalmpBNLgnjASDsY3eT19phIRSPxoCcx8TgaCBpSjLSxfLuW+BT2qIA9jvQwCNL77DHl2RkcL/FF5ttVp+ZMBRfBLaAKCjV8+7vXj3DCidCYIaW6rhNrL0VSU8xIVuklisQIj9CAdA0KxIny0uldGTwxTh+GkTRPaDh1/06kiCs14YHpzLdU87XcXFbrJjq88lIq4kQTgWcfhQmDOoJ5SLBPJcGaTQwgLKnZFeIhkghrE2XFhODOn7wIrfOaa/juolq/LuIogyNwDE6BCy5BHdyCBmgCDJ7AC3gD79az9Wp9WJ+z1pJVzByCf7K+fgH9+Z/T</latexit><latexit sha1_base64="KAAe+IfS96EJriixWDJslYBAwtI=">AAACF3icbZDLSsNAFIYn9VbrLerSzWARXEhJRNBl0Y3LKvYCbQyT6aQdOjMJM5NCCXkLN76KGxeKuNWdb+OkzULb/jDw8Z9zmHP+IGZUacf5sUorq2vrG+XNytb2zu6evX/QUlEiMWniiEWyEyBFGBWkqalmpBNLgnjASDsY3eT19phIRSPxoCcx8TgaCBpSjLSxfLuW+BT2qIA9jvQwCNL77DHl2RkcL/FF5ttVp+ZMBRfBLaAKCjV8+7vXj3DCidCYIaW6rhNrL0VSU8xIVuklisQIj9CAdA0KxIny0uldGTwxTh+GkTRPaDh1/06kiCs14YHpzLdU87XcXFbrJjq88lIq4kQTgWcfhQmDOoJ5SLBPJcGaTQwgLKnZFeIhkghrE2XFhODOn7wIrfOaa/juolq/LuIogyNwDE6BCy5BHdyCBmgCDJ7AC3gD79az9Wp9WJ+z1pJVzByCf7K+fgH9+Z/T</latexit><latexit sha1_base64="KAAe+IfS96EJriixWDJslYBAwtI=">AAACF3icbZDLSsNAFIYn9VbrLerSzWARXEhJRNBl0Y3LKvYCbQyT6aQdOjMJM5NCCXkLN76KGxeKuNWdb+OkzULb/jDw8Z9zmHP+IGZUacf5sUorq2vrG+XNytb2zu6evX/QUlEiMWniiEWyEyBFGBWkqalmpBNLgnjASDsY3eT19phIRSPxoCcx8TgaCBpSjLSxfLuW+BT2qIA9jvQwCNL77DHl2RkcL/FF5ttVp+ZMBRfBLaAKCjV8+7vXj3DCidCYIaW6rhNrL0VSU8xIVuklisQIj9CAdA0KxIny0uldGTwxTh+GkTRPaDh1/06kiCs14YHpzLdU87XcXFbrJjq88lIq4kQTgWcfhQmDOoJ5SLBPJcGaTQwgLKnZFeIhkghrE2XFhODOn7wIrfOaa/juolq/LuIogyNwDE6BCy5BHdyCBmgCDJ7AC3gD79az9Wp9WJ+z1pJVzByCf7K+fgH9+Z/T</latexit>

- contains singular values where ⌃ = diag(�1, . . . ,�r)<latexit sha1_base64="7m8/hqoL4MJwUt2j/j35KcIRxHw=">AAACGnicbVDLSgMxFM34rPU16tJNsAgVpMyIoBuh6MZlRfuAzlAyaWYamnmQ3BHL0O9w46+4caGIO3Hj35hpZ6GtB0IO59x7k3u8RHAFlvVtLCwuLa+sltbK6xubW9vmzm5LxamkrEljEcuORxQTPGJN4CBYJ5GMhJ5gbW94lfvteyYVj6M7GCXMDUkQcZ9TAlrqmbZzy4OQ4AvsAHsAgKzPSTCuOiqXe/YxdvoxKH1NBXnUMytWzZoAzxO7IBVUoNEzP/UImoYsAiqIUl3bSsDNiAROBRuXnVSxhNAhCVhX04iETLnZZLUxPtRKH/ux1CcCPFF/d2QkVGoUeroyJDBQs14u/ud1U/DP3YxHSQosotOH/FRgiHGeE+5zySiIkSaESq7/iumASEJBp1nWIdizK8+T1knN1vzmtFK/LOIooX10gKrIRmeojq5RAzURRY/oGb2iN+PJeDHejY9p6YJR9OyhPzC+fgD6JqAv</latexit><latexit sha1_base64="7m8/hqoL4MJwUt2j/j35KcIRxHw=">AAACGnicbVDLSgMxFM34rPU16tJNsAgVpMyIoBuh6MZlRfuAzlAyaWYamnmQ3BHL0O9w46+4caGIO3Hj35hpZ6GtB0IO59x7k3u8RHAFlvVtLCwuLa+sltbK6xubW9vmzm5LxamkrEljEcuORxQTPGJN4CBYJ5GMhJ5gbW94lfvteyYVj6M7GCXMDUkQcZ9TAlrqmbZzy4OQ4AvsAHsAgKzPSTCuOiqXe/YxdvoxKH1NBXnUMytWzZoAzxO7IBVUoNEzP/UImoYsAiqIUl3bSsDNiAROBRuXnVSxhNAhCVhX04iETLnZZLUxPtRKH/ux1CcCPFF/d2QkVGoUeroyJDBQs14u/ud1U/DP3YxHSQosotOH/FRgiHGeE+5zySiIkSaESq7/iumASEJBp1nWIdizK8+T1knN1vzmtFK/LOIooX10gKrIRmeojq5RAzURRY/oGb2iN+PJeDHejY9p6YJR9OyhPzC+fgD6JqAv</latexit><latexit sha1_base64="7m8/hqoL4MJwUt2j/j35KcIRxHw=">AAACGnicbVDLSgMxFM34rPU16tJNsAgVpMyIoBuh6MZlRfuAzlAyaWYamnmQ3BHL0O9w46+4caGIO3Hj35hpZ6GtB0IO59x7k3u8RHAFlvVtLCwuLa+sltbK6xubW9vmzm5LxamkrEljEcuORxQTPGJN4CBYJ5GMhJ5gbW94lfvteyYVj6M7GCXMDUkQcZ9TAlrqmbZzy4OQ4AvsAHsAgKzPSTCuOiqXe/YxdvoxKH1NBXnUMytWzZoAzxO7IBVUoNEzP/UImoYsAiqIUl3bSsDNiAROBRuXnVSxhNAhCVhX04iETLnZZLUxPtRKH/ux1CcCPFF/d2QkVGoUeroyJDBQs14u/ud1U/DP3YxHSQosotOH/FRgiHGeE+5zySiIkSaESq7/iumASEJBp1nWIdizK8+T1knN1vzmtFK/LOIooX10gKrIRmeojq5RAzURRY/oGb2iN+PJeDHejY9p6YJR9OyhPzC+fgD6JqAv</latexit><latexit sha1_base64="7m8/hqoL4MJwUt2j/j35KcIRxHw=">AAACGnicbVDLSgMxFM34rPU16tJNsAgVpMyIoBuh6MZlRfuAzlAyaWYamnmQ3BHL0O9w46+4caGIO3Hj35hpZ6GtB0IO59x7k3u8RHAFlvVtLCwuLa+sltbK6xubW9vmzm5LxamkrEljEcuORxQTPGJN4CBYJ5GMhJ5gbW94lfvteyYVj6M7GCXMDUkQcZ9TAlrqmbZzy4OQ4AvsAHsAgKzPSTCuOiqXe/YxdvoxKH1NBXnUMytWzZoAzxO7IBVUoNEzP/UImoYsAiqIUl3bSsDNiAROBRuXnVSxhNAhCVhX04iETLnZZLUxPtRKH/ux1CcCPFF/d2QkVGoUeroyJDBQs14u/ud1U/DP3YxHSQosotOH/FRgiHGeE+5zySiIkSaESq7/iumASEJBp1nWIdizK8+T1knN1vzmtFK/LOIooX10gKrIRmeojq5RAzURRY/oGb2iN+PJeDHejY9p6YJR9OyhPzC+fgD6JqAv</latexit>

�1 � �2 � · · · � �r<latexit sha1_base64="/CxJoGcpgoJimdf22nG+xk/jFt0=">AAACGHicbVC7TsMwFL3hWcorwMhiUSExlaRCgrGChbFI9CG1UeS4TmrVToLtIFVRP4OFX2FhACHWbvwNbpuBthzJ0rnn3Gv7niDlTGnH+bHW1jc2t7ZLO+Xdvf2DQ/vouKWSTBLaJAlPZCfAinIW06ZmmtNOKikWAaftYHg39dvPVCqWxI96lFJP4ChmISNYG8m3L3uKRQL7LupF9AkVVa2o+olWC4b07YpTdWZAq8QtSAUKNHx7Ym4hmaCxJhwr1XWdVHs5lpoRTsflXqZoiskQR7RraIwFVV4+W2yMzo3SR2EizYk1mql/J3IslBqJwHQKrAdq2ZuK/3ndTIc3Xs7iNNM0JvOHwowjnaBpSqjPJCWajwzBRDLzV0QGWGKiTZZlE4K7vPIqadWqruEPV5X6bRFHCU7hDC7AhWuowz00oAkEXuANPuDTerXerS/re966ZhUzJ7AAa/ILYBCfWw==</latexit><latexit sha1_base64="/CxJoGcpgoJimdf22nG+xk/jFt0=">AAACGHicbVC7TsMwFL3hWcorwMhiUSExlaRCgrGChbFI9CG1UeS4TmrVToLtIFVRP4OFX2FhACHWbvwNbpuBthzJ0rnn3Gv7niDlTGnH+bHW1jc2t7ZLO+Xdvf2DQ/vouKWSTBLaJAlPZCfAinIW06ZmmtNOKikWAaftYHg39dvPVCqWxI96lFJP4ChmISNYG8m3L3uKRQL7LupF9AkVVa2o+olWC4b07YpTdWZAq8QtSAUKNHx7Ym4hmaCxJhwr1XWdVHs5lpoRTsflXqZoiskQR7RraIwFVV4+W2yMzo3SR2EizYk1mql/J3IslBqJwHQKrAdq2ZuK/3ndTIc3Xs7iNNM0JvOHwowjnaBpSqjPJCWajwzBRDLzV0QGWGKiTZZlE4K7vPIqadWqruEPV5X6bRFHCU7hDC7AhWuowz00oAkEXuANPuDTerXerS/re966ZhUzJ7AAa/ILYBCfWw==</latexit><latexit sha1_base64="/CxJoGcpgoJimdf22nG+xk/jFt0=">AAACGHicbVC7TsMwFL3hWcorwMhiUSExlaRCgrGChbFI9CG1UeS4TmrVToLtIFVRP4OFX2FhACHWbvwNbpuBthzJ0rnn3Gv7niDlTGnH+bHW1jc2t7ZLO+Xdvf2DQ/vouKWSTBLaJAlPZCfAinIW06ZmmtNOKikWAaftYHg39dvPVCqWxI96lFJP4ChmISNYG8m3L3uKRQL7LupF9AkVVa2o+olWC4b07YpTdWZAq8QtSAUKNHx7Ym4hmaCxJhwr1XWdVHs5lpoRTsflXqZoiskQR7RraIwFVV4+W2yMzo3SR2EizYk1mql/J3IslBqJwHQKrAdq2ZuK/3ndTIc3Xs7iNNM0JvOHwowjnaBpSqjPJCWajwzBRDLzV0QGWGKiTZZlE4K7vPIqadWqruEPV5X6bRFHCU7hDC7AhWuowz00oAkEXuANPuDTerXerS/re966ZhUzJ7AAa/ILYBCfWw==</latexit><latexit sha1_base64="/CxJoGcpgoJimdf22nG+xk/jFt0=">AAACGHicbVC7TsMwFL3hWcorwMhiUSExlaRCgrGChbFI9CG1UeS4TmrVToLtIFVRP4OFX2FhACHWbvwNbpuBthzJ0rnn3Gv7niDlTGnH+bHW1jc2t7ZLO+Xdvf2DQ/vouKWSTBLaJAlPZCfAinIW06ZmmtNOKikWAaftYHg39dvPVCqWxI96lFJP4ChmISNYG8m3L3uKRQL7LupF9AkVVa2o+olWC4b07YpTdWZAq8QtSAUKNHx7Ym4hmaCxJhwr1XWdVHs5lpoRTsflXqZoiskQR7RraIwFVV4+W2yMzo3SR2EizYk1mql/J3IslBqJwHQKrAdq2ZuK/3ndTIc3Xs7iNNM0JvOHwowjnaBpSqjPJCWajwzBRDLzV0QGWGKiTZZlE4K7vPIqadWqruEPV5X6bRFHCU7hDC7AhWuowz00oAkEXuANPuDTerXerS/re966ZhUzJ7AAa/ILYBCfWw==</latexit>

Page 137: COMP 414/514: Optimization - Algorithms, Complexity and

Matrices

- Matrix singular value decomposition:

r {m,n}<latexit sha1_base64="RyyT3zeYAEA4tcZj4lBVYOs9IVw=">AAAB+HicbZDLSgMxFIbPeK310lGXboJFcCFlRgRdFt24rGAv0BlKJs20oUlmTDJCHfokblwo4tZHcefbmLaz0NYfAh//OYdz8kcpZ9p43rezsrq2vrFZ2ipv7+zuVdz9g5ZOMkVokyQ8UZ0Ia8qZpE3DDKedVFEsIk7b0ehmWm8/UqVZIu/NOKWhwAPJYkawsVbPrSgUcPqAglycIRlMem7Vq3kzoWXwC6hCoUbP/Qr6CckElYZwrHXX91IT5lgZRjidlINM0xSTER7QrkWJBdVhPjt8gk6s00dxouyTBs3c3xM5FlqPRWQ7BTZDvVibmv/VupmJr8KcyTQzVJL5ojjjyCRomgLqM0WJ4WMLmChmb0VkiBUmxmZVtiH4i19ehtZ5zbd8d1GtXxdxlOAIjuEUfLiEOtxCA5pAIINneIU358l5cd6dj3nrilPMHMIfOZ8/oHqSaA==</latexit><latexit sha1_base64="RyyT3zeYAEA4tcZj4lBVYOs9IVw=">AAAB+HicbZDLSgMxFIbPeK310lGXboJFcCFlRgRdFt24rGAv0BlKJs20oUlmTDJCHfokblwo4tZHcefbmLaz0NYfAh//OYdz8kcpZ9p43rezsrq2vrFZ2ipv7+zuVdz9g5ZOMkVokyQ8UZ0Ia8qZpE3DDKedVFEsIk7b0ehmWm8/UqVZIu/NOKWhwAPJYkawsVbPrSgUcPqAglycIRlMem7Vq3kzoWXwC6hCoUbP/Qr6CckElYZwrHXX91IT5lgZRjidlINM0xSTER7QrkWJBdVhPjt8gk6s00dxouyTBs3c3xM5FlqPRWQ7BTZDvVibmv/VupmJr8KcyTQzVJL5ojjjyCRomgLqM0WJ4WMLmChmb0VkiBUmxmZVtiH4i19ehtZ5zbd8d1GtXxdxlOAIjuEUfLiEOtxCA5pAIINneIU358l5cd6dj3nrilPMHMIfOZ8/oHqSaA==</latexit><latexit sha1_base64="RyyT3zeYAEA4tcZj4lBVYOs9IVw=">AAAB+HicbZDLSgMxFIbPeK310lGXboJFcCFlRgRdFt24rGAv0BlKJs20oUlmTDJCHfokblwo4tZHcefbmLaz0NYfAh//OYdz8kcpZ9p43rezsrq2vrFZ2ipv7+zuVdz9g5ZOMkVokyQ8UZ0Ia8qZpE3DDKedVFEsIk7b0ehmWm8/UqVZIu/NOKWhwAPJYkawsVbPrSgUcPqAglycIRlMem7Vq3kzoWXwC6hCoUbP/Qr6CckElYZwrHXX91IT5lgZRjidlINM0xSTER7QrkWJBdVhPjt8gk6s00dxouyTBs3c3xM5FlqPRWQ7BTZDvVibmv/VupmJr8KcyTQzVJL5ojjjyCRomgLqM0WJ4WMLmChmb0VkiBUmxmZVtiH4i19ehtZ5zbd8d1GtXxdxlOAIjuEUfLiEOtxCA5pAIINneIU358l5cd6dj3nrilPMHMIfOZ8/oHqSaA==</latexit><latexit sha1_base64="RyyT3zeYAEA4tcZj4lBVYOs9IVw=">AAAB+HicbZDLSgMxFIbPeK310lGXboJFcCFlRgRdFt24rGAv0BlKJs20oUlmTDJCHfokblwo4tZHcefbmLaz0NYfAh//OYdz8kcpZ9p43rezsrq2vrFZ2ipv7+zuVdz9g5ZOMkVokyQ8UZ0Ia8qZpE3DDKedVFEsIk7b0ehmWm8/UqVZIu/NOKWhwAPJYkawsVbPrSgUcPqAglycIRlMem7Vq3kzoWXwC6hCoUbP/Qr6CckElYZwrHXX91IT5lgZRjidlINM0xSTER7QrkWJBdVhPjt8gk6s00dxouyTBs3c3xM5FlqPRWQ7BTZDvVibmv/VupmJr8KcyTQzVJL5ojjjyCRomgLqM0WJ4WMLmChmb0VkiBUmxmZVtiH4i19ehtZ5zbd8d1GtXxdxlOAIjuEUfLiEOtxCA5pAIINneIU358l5cd6dj3nrilPMHMIfOZ8/oHqSaA==</latexit>

A 2 Rm⇥n<latexit sha1_base64="MxwCSL/gyMr0pbYAXqcODXgj2Pw=">AAACBnicbVBNS8NAEJ34WetX1KMIi0XwVBIR9Fj14rGK/YAmls120y7dbMLuRighJy/+FS8eFPHqb/Dmv3HT9qCtDwYe780wMy9IOFPacb6thcWl5ZXV0lp5fWNza9ve2W2qOJWENkjMY9kOsKKcCdrQTHPaTiTFUcBpKxheFX7rgUrFYnGnRwn1I9wXLGQEayN17YML5DGBvAjrQRBkt/l9FiFPs4gqJPKuXXGqzhhonrhTUoEp6l37y+vFJI2o0IRjpTquk2g/w1Izwmle9lJFE0yGuE87hgps9vjZ+I0cHRmlh8JYmhIajdXfExmOlBpFgekszlWzXiH+53VSHZ77GRNJqqkgk0VhypGOUZEJ6jFJieYjQzCRzNyKyABLTLRJrmxCcGdfnifNk6pr+M1ppXY5jaME+3AIx+DCGdTgGurQAAKP8Ayv8GY9WS/Wu/UxaV2wpjN78AfW5w+D2JiE</latexit><latexit sha1_base64="MxwCSL/gyMr0pbYAXqcODXgj2Pw=">AAACBnicbVBNS8NAEJ34WetX1KMIi0XwVBIR9Fj14rGK/YAmls120y7dbMLuRighJy/+FS8eFPHqb/Dmv3HT9qCtDwYe780wMy9IOFPacb6thcWl5ZXV0lp5fWNza9ve2W2qOJWENkjMY9kOsKKcCdrQTHPaTiTFUcBpKxheFX7rgUrFYnGnRwn1I9wXLGQEayN17YML5DGBvAjrQRBkt/l9FiFPs4gqJPKuXXGqzhhonrhTUoEp6l37y+vFJI2o0IRjpTquk2g/w1Izwmle9lJFE0yGuE87hgps9vjZ+I0cHRmlh8JYmhIajdXfExmOlBpFgekszlWzXiH+53VSHZ77GRNJqqkgk0VhypGOUZEJ6jFJieYjQzCRzNyKyABLTLRJrmxCcGdfnifNk6pr+M1ppXY5jaME+3AIx+DCGdTgGurQAAKP8Ayv8GY9WS/Wu/UxaV2wpjN78AfW5w+D2JiE</latexit><latexit sha1_base64="MxwCSL/gyMr0pbYAXqcODXgj2Pw=">AAACBnicbVBNS8NAEJ34WetX1KMIi0XwVBIR9Fj14rGK/YAmls120y7dbMLuRighJy/+FS8eFPHqb/Dmv3HT9qCtDwYe780wMy9IOFPacb6thcWl5ZXV0lp5fWNza9ve2W2qOJWENkjMY9kOsKKcCdrQTHPaTiTFUcBpKxheFX7rgUrFYnGnRwn1I9wXLGQEayN17YML5DGBvAjrQRBkt/l9FiFPs4gqJPKuXXGqzhhonrhTUoEp6l37y+vFJI2o0IRjpTquk2g/w1Izwmle9lJFE0yGuE87hgps9vjZ+I0cHRmlh8JYmhIajdXfExmOlBpFgekszlWzXiH+53VSHZ77GRNJqqkgk0VhypGOUZEJ6jFJieYjQzCRzNyKyABLTLRJrmxCcGdfnifNk6pr+M1ppXY5jaME+3AIx+DCGdTgGurQAAKP8Ayv8GY9WS/Wu/UxaV2wpjN78AfW5w+D2JiE</latexit><latexit sha1_base64="MxwCSL/gyMr0pbYAXqcODXgj2Pw=">AAACBnicbVBNS8NAEJ34WetX1KMIi0XwVBIR9Fj14rGK/YAmls120y7dbMLuRighJy/+FS8eFPHqb/Dmv3HT9qCtDwYe780wMy9IOFPacb6thcWl5ZXV0lp5fWNza9ve2W2qOJWENkjMY9kOsKKcCdrQTHPaTiTFUcBpKxheFX7rgUrFYnGnRwn1I9wXLGQEayN17YML5DGBvAjrQRBkt/l9FiFPs4gqJPKuXXGqzhhonrhTUoEp6l37y+vFJI2o0IRjpTquk2g/w1Izwmle9lJFE0yGuE87hgps9vjZ+I0cHRmlh8JYmhIajdXfExmOlBpFgekszlWzXiH+53VSHZ77GRNJqqkgk0VhypGOUZEJ6jFJieYjQzCRzNyKyABLTLRJrmxCcGdfnifNk6pr+M1ppXY5jaME+3AIx+DCGdTgGurQAAKP8Ayv8GY9WS/Wu/UxaV2wpjN78AfW5w+D2JiE</latexit>

A = U⌃V > =rX

i=1

�iuiv>i , U 2 Rm⇥r,⌃ 2 Rr⇥r, V 2 Rn⇥r

<latexit sha1_base64="lKyZjiBFpx910sCGQFpkkzLMSmw=">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</latexit><latexit sha1_base64="lKyZjiBFpx910sCGQFpkkzLMSmw=">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</latexit><latexit sha1_base64="lKyZjiBFpx910sCGQFpkkzLMSmw=">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</latexit><latexit sha1_base64="lKyZjiBFpx910sCGQFpkkzLMSmw=">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</latexit>

- rank(A) = r min{m,n}<latexit sha1_base64="L/IAKAyqFfjh9StE51AH7q6v9V4=">AAACDXicbZBNSwMxEIazflu/qh69BKtQQWRXBL0IVS8eFWwVuqVk02kbmmTXZFYsS/+AF/+KFw+KePXuzX9j+nHQ6guBh3dmmMwbJVJY9P0vb2Jyanpmdm4+t7C4tLySX12r2Dg1HMo8lrG5iZgFKTSUUaCEm8QAU5GE66hz1q9f34GxItZX2E2gplhLi6bgDJ1Vz2+FCPeYGaY7veLJDj2mhoYSbmmohA4ztUt12KvnC/6ePxD9C8EICmSki3r+M2zEPFWgkUtmbTXwE6xlzKDgEnq5MLWQMN5hLag61EyBrWWDa3p02zkN2oyNexrpwP05kTFlbVdFrlMxbNvxWt/8r1ZNsXlUy4ROUgTNh4uaqaQY0340tCEMcJRdB4wb4f5KeZsZxtEFmHMhBOMn/4XK/l7g+PKgUDodxTFHNsgmKZKAHJISOScXpEw4eSBP5IW8eo/es/fmvQ9bJ7zRzDr5Je/jG7h8mq4=</latexit><latexit sha1_base64="L/IAKAyqFfjh9StE51AH7q6v9V4=">AAACDXicbZBNSwMxEIazflu/qh69BKtQQWRXBL0IVS8eFWwVuqVk02kbmmTXZFYsS/+AF/+KFw+KePXuzX9j+nHQ6guBh3dmmMwbJVJY9P0vb2Jyanpmdm4+t7C4tLySX12r2Dg1HMo8lrG5iZgFKTSUUaCEm8QAU5GE66hz1q9f34GxItZX2E2gplhLi6bgDJ1Vz2+FCPeYGaY7veLJDj2mhoYSbmmohA4ztUt12KvnC/6ePxD9C8EICmSki3r+M2zEPFWgkUtmbTXwE6xlzKDgEnq5MLWQMN5hLag61EyBrWWDa3p02zkN2oyNexrpwP05kTFlbVdFrlMxbNvxWt/8r1ZNsXlUy4ROUgTNh4uaqaQY0340tCEMcJRdB4wb4f5KeZsZxtEFmHMhBOMn/4XK/l7g+PKgUDodxTFHNsgmKZKAHJISOScXpEw4eSBP5IW8eo/es/fmvQ9bJ7zRzDr5Je/jG7h8mq4=</latexit><latexit sha1_base64="L/IAKAyqFfjh9StE51AH7q6v9V4=">AAACDXicbZBNSwMxEIazflu/qh69BKtQQWRXBL0IVS8eFWwVuqVk02kbmmTXZFYsS/+AF/+KFw+KePXuzX9j+nHQ6guBh3dmmMwbJVJY9P0vb2Jyanpmdm4+t7C4tLySX12r2Dg1HMo8lrG5iZgFKTSUUaCEm8QAU5GE66hz1q9f34GxItZX2E2gplhLi6bgDJ1Vz2+FCPeYGaY7veLJDj2mhoYSbmmohA4ztUt12KvnC/6ePxD9C8EICmSki3r+M2zEPFWgkUtmbTXwE6xlzKDgEnq5MLWQMN5hLag61EyBrWWDa3p02zkN2oyNexrpwP05kTFlbVdFrlMxbNvxWt/8r1ZNsXlUy4ROUgTNh4uaqaQY0340tCEMcJRdB4wb4f5KeZsZxtEFmHMhBOMn/4XK/l7g+PKgUDodxTFHNsgmKZKAHJISOScXpEw4eSBP5IW8eo/es/fmvQ9bJ7zRzDr5Je/jG7h8mq4=</latexit><latexit sha1_base64="L/IAKAyqFfjh9StE51AH7q6v9V4=">AAACDXicbZBNSwMxEIazflu/qh69BKtQQWRXBL0IVS8eFWwVuqVk02kbmmTXZFYsS/+AF/+KFw+KePXuzX9j+nHQ6guBh3dmmMwbJVJY9P0vb2Jyanpmdm4+t7C4tLySX12r2Dg1HMo8lrG5iZgFKTSUUaCEm8QAU5GE66hz1q9f34GxItZX2E2gplhLi6bgDJ1Vz2+FCPeYGaY7veLJDj2mhoYSbmmohA4ztUt12KvnC/6ePxD9C8EICmSki3r+M2zEPFWgkUtmbTXwE6xlzKDgEnq5MLWQMN5hLag61EyBrWWDa3p02zkN2oyNexrpwP05kTFlbVdFrlMxbNvxWt/8r1ZNsXlUy4ROUgTNh4uaqaQY0340tCEMcJRdB4wb4f5KeZsZxtEFmHMhBOMn/4XK/l7g+PKgUDodxTFHNsgmKZKAHJISOScXpEw4eSBP5IW8eo/es/fmvQ9bJ7zRzDr5Je/jG7h8mq4=</latexit>

- are the left and right singular vectorsui 2 Rm, vi 2 Rn<latexit sha1_base64="KAAe+IfS96EJriixWDJslYBAwtI=">AAACF3icbZDLSsNAFIYn9VbrLerSzWARXEhJRNBl0Y3LKvYCbQyT6aQdOjMJM5NCCXkLN76KGxeKuNWdb+OkzULb/jDw8Z9zmHP+IGZUacf5sUorq2vrG+XNytb2zu6evX/QUlEiMWniiEWyEyBFGBWkqalmpBNLgnjASDsY3eT19phIRSPxoCcx8TgaCBpSjLSxfLuW+BT2qIA9jvQwCNL77DHl2RkcL/FF5ttVp+ZMBRfBLaAKCjV8+7vXj3DCidCYIaW6rhNrL0VSU8xIVuklisQIj9CAdA0KxIny0uldGTwxTh+GkTRPaDh1/06kiCs14YHpzLdU87XcXFbrJjq88lIq4kQTgWcfhQmDOoJ5SLBPJcGaTQwgLKnZFeIhkghrE2XFhODOn7wIrfOaa/juolq/LuIogyNwDE6BCy5BHdyCBmgCDJ7AC3gD79az9Wp9WJ+z1pJVzByCf7K+fgH9+Z/T</latexit><latexit sha1_base64="KAAe+IfS96EJriixWDJslYBAwtI=">AAACF3icbZDLSsNAFIYn9VbrLerSzWARXEhJRNBl0Y3LKvYCbQyT6aQdOjMJM5NCCXkLN76KGxeKuNWdb+OkzULb/jDw8Z9zmHP+IGZUacf5sUorq2vrG+XNytb2zu6evX/QUlEiMWniiEWyEyBFGBWkqalmpBNLgnjASDsY3eT19phIRSPxoCcx8TgaCBpSjLSxfLuW+BT2qIA9jvQwCNL77DHl2RkcL/FF5ttVp+ZMBRfBLaAKCjV8+7vXj3DCidCYIaW6rhNrL0VSU8xIVuklisQIj9CAdA0KxIny0uldGTwxTh+GkTRPaDh1/06kiCs14YHpzLdU87XcXFbrJjq88lIq4kQTgWcfhQmDOoJ5SLBPJcGaTQwgLKnZFeIhkghrE2XFhODOn7wIrfOaa/juolq/LuIogyNwDE6BCy5BHdyCBmgCDJ7AC3gD79az9Wp9WJ+z1pJVzByCf7K+fgH9+Z/T</latexit><latexit sha1_base64="KAAe+IfS96EJriixWDJslYBAwtI=">AAACF3icbZDLSsNAFIYn9VbrLerSzWARXEhJRNBl0Y3LKvYCbQyT6aQdOjMJM5NCCXkLN76KGxeKuNWdb+OkzULb/jDw8Z9zmHP+IGZUacf5sUorq2vrG+XNytb2zu6evX/QUlEiMWniiEWyEyBFGBWkqalmpBNLgnjASDsY3eT19phIRSPxoCcx8TgaCBpSjLSxfLuW+BT2qIA9jvQwCNL77DHl2RkcL/FF5ttVp+ZMBRfBLaAKCjV8+7vXj3DCidCYIaW6rhNrL0VSU8xIVuklisQIj9CAdA0KxIny0uldGTwxTh+GkTRPaDh1/06kiCs14YHpzLdU87XcXFbrJjq88lIq4kQTgWcfhQmDOoJ5SLBPJcGaTQwgLKnZFeIhkghrE2XFhODOn7wIrfOaa/juolq/LuIogyNwDE6BCy5BHdyCBmgCDJ7AC3gD79az9Wp9WJ+z1pJVzByCf7K+fgH9+Z/T</latexit><latexit sha1_base64="KAAe+IfS96EJriixWDJslYBAwtI=">AAACF3icbZDLSsNAFIYn9VbrLerSzWARXEhJRNBl0Y3LKvYCbQyT6aQdOjMJM5NCCXkLN76KGxeKuNWdb+OkzULb/jDw8Z9zmHP+IGZUacf5sUorq2vrG+XNytb2zu6evX/QUlEiMWniiEWyEyBFGBWkqalmpBNLgnjASDsY3eT19phIRSPxoCcx8TgaCBpSjLSxfLuW+BT2qIA9jvQwCNL77DHl2RkcL/FF5ttVp+ZMBRfBLaAKCjV8+7vXj3DCidCYIaW6rhNrL0VSU8xIVuklisQIj9CAdA0KxIny0uldGTwxTh+GkTRPaDh1/06kiCs14YHpzLdU87XcXFbrJjq88lIq4kQTgWcfhQmDOoJ5SLBPJcGaTQwgLKnZFeIhkghrE2XFhODOn7wIrfOaa/juolq/LuIogyNwDE6BCy5BHdyCBmgCDJ7AC3gD79az9Wp9WJ+z1pJVzByCf7K+fgH9+Z/T</latexit>

- contains singular values where ⌃ = diag(�1, . . . ,�r)<latexit sha1_base64="7m8/hqoL4MJwUt2j/j35KcIRxHw=">AAACGnicbVDLSgMxFM34rPU16tJNsAgVpMyIoBuh6MZlRfuAzlAyaWYamnmQ3BHL0O9w46+4caGIO3Hj35hpZ6GtB0IO59x7k3u8RHAFlvVtLCwuLa+sltbK6xubW9vmzm5LxamkrEljEcuORxQTPGJN4CBYJ5GMhJ5gbW94lfvteyYVj6M7GCXMDUkQcZ9TAlrqmbZzy4OQ4AvsAHsAgKzPSTCuOiqXe/YxdvoxKH1NBXnUMytWzZoAzxO7IBVUoNEzP/UImoYsAiqIUl3bSsDNiAROBRuXnVSxhNAhCVhX04iETLnZZLUxPtRKH/ux1CcCPFF/d2QkVGoUeroyJDBQs14u/ud1U/DP3YxHSQosotOH/FRgiHGeE+5zySiIkSaESq7/iumASEJBp1nWIdizK8+T1knN1vzmtFK/LOIooX10gKrIRmeojq5RAzURRY/oGb2iN+PJeDHejY9p6YJR9OyhPzC+fgD6JqAv</latexit><latexit sha1_base64="7m8/hqoL4MJwUt2j/j35KcIRxHw=">AAACGnicbVDLSgMxFM34rPU16tJNsAgVpMyIoBuh6MZlRfuAzlAyaWYamnmQ3BHL0O9w46+4caGIO3Hj35hpZ6GtB0IO59x7k3u8RHAFlvVtLCwuLa+sltbK6xubW9vmzm5LxamkrEljEcuORxQTPGJN4CBYJ5GMhJ5gbW94lfvteyYVj6M7GCXMDUkQcZ9TAlrqmbZzy4OQ4AvsAHsAgKzPSTCuOiqXe/YxdvoxKH1NBXnUMytWzZoAzxO7IBVUoNEzP/UImoYsAiqIUl3bSsDNiAROBRuXnVSxhNAhCVhX04iETLnZZLUxPtRKH/ux1CcCPFF/d2QkVGoUeroyJDBQs14u/ud1U/DP3YxHSQosotOH/FRgiHGeE+5zySiIkSaESq7/iumASEJBp1nWIdizK8+T1knN1vzmtFK/LOIooX10gKrIRmeojq5RAzURRY/oGb2iN+PJeDHejY9p6YJR9OyhPzC+fgD6JqAv</latexit><latexit sha1_base64="7m8/hqoL4MJwUt2j/j35KcIRxHw=">AAACGnicbVDLSgMxFM34rPU16tJNsAgVpMyIoBuh6MZlRfuAzlAyaWYamnmQ3BHL0O9w46+4caGIO3Hj35hpZ6GtB0IO59x7k3u8RHAFlvVtLCwuLa+sltbK6xubW9vmzm5LxamkrEljEcuORxQTPGJN4CBYJ5GMhJ5gbW94lfvteyYVj6M7GCXMDUkQcZ9TAlrqmbZzy4OQ4AvsAHsAgKzPSTCuOiqXe/YxdvoxKH1NBXnUMytWzZoAzxO7IBVUoNEzP/UImoYsAiqIUl3bSsDNiAROBRuXnVSxhNAhCVhX04iETLnZZLUxPtRKH/ux1CcCPFF/d2QkVGoUeroyJDBQs14u/ud1U/DP3YxHSQosotOH/FRgiHGeE+5zySiIkSaESq7/iumASEJBp1nWIdizK8+T1knN1vzmtFK/LOIooX10gKrIRmeojq5RAzURRY/oGb2iN+PJeDHejY9p6YJR9OyhPzC+fgD6JqAv</latexit><latexit sha1_base64="7m8/hqoL4MJwUt2j/j35KcIRxHw=">AAACGnicbVDLSgMxFM34rPU16tJNsAgVpMyIoBuh6MZlRfuAzlAyaWYamnmQ3BHL0O9w46+4caGIO3Hj35hpZ6GtB0IO59x7k3u8RHAFlvVtLCwuLa+sltbK6xubW9vmzm5LxamkrEljEcuORxQTPGJN4CBYJ5GMhJ5gbW94lfvteyYVj6M7GCXMDUkQcZ9TAlrqmbZzy4OQ4AvsAHsAgKzPSTCuOiqXe/YxdvoxKH1NBXnUMytWzZoAzxO7IBVUoNEzP/UImoYsAiqIUl3bSsDNiAROBRuXnVSxhNAhCVhX04iETLnZZLUxPtRKH/ux1CcCPFF/d2QkVGoUeroyJDBQs14u/ud1U/DP3YxHSQosotOH/FRgiHGeE+5zySiIkSaESq7/iumASEJBp1nWIdizK8+T1knN1vzmtFK/LOIooX10gKrIRmeojq5RAzURRY/oGb2iN+PJeDHejY9p6YJR9OyhPzC+fgD6JqAv</latexit>

�1 � �2 � · · · � �r<latexit sha1_base64="/CxJoGcpgoJimdf22nG+xk/jFt0=">AAACGHicbVC7TsMwFL3hWcorwMhiUSExlaRCgrGChbFI9CG1UeS4TmrVToLtIFVRP4OFX2FhACHWbvwNbpuBthzJ0rnn3Gv7niDlTGnH+bHW1jc2t7ZLO+Xdvf2DQ/vouKWSTBLaJAlPZCfAinIW06ZmmtNOKikWAaftYHg39dvPVCqWxI96lFJP4ChmISNYG8m3L3uKRQL7LupF9AkVVa2o+olWC4b07YpTdWZAq8QtSAUKNHx7Ym4hmaCxJhwr1XWdVHs5lpoRTsflXqZoiskQR7RraIwFVV4+W2yMzo3SR2EizYk1mql/J3IslBqJwHQKrAdq2ZuK/3ndTIc3Xs7iNNM0JvOHwowjnaBpSqjPJCWajwzBRDLzV0QGWGKiTZZlE4K7vPIqadWqruEPV5X6bRFHCU7hDC7AhWuowz00oAkEXuANPuDTerXerS/re966ZhUzJ7AAa/ILYBCfWw==</latexit><latexit sha1_base64="/CxJoGcpgoJimdf22nG+xk/jFt0=">AAACGHicbVC7TsMwFL3hWcorwMhiUSExlaRCgrGChbFI9CG1UeS4TmrVToLtIFVRP4OFX2FhACHWbvwNbpuBthzJ0rnn3Gv7niDlTGnH+bHW1jc2t7ZLO+Xdvf2DQ/vouKWSTBLaJAlPZCfAinIW06ZmmtNOKikWAaftYHg39dvPVCqWxI96lFJP4ChmISNYG8m3L3uKRQL7LupF9AkVVa2o+olWC4b07YpTdWZAq8QtSAUKNHx7Ym4hmaCxJhwr1XWdVHs5lpoRTsflXqZoiskQR7RraIwFVV4+W2yMzo3SR2EizYk1mql/J3IslBqJwHQKrAdq2ZuK/3ndTIc3Xs7iNNM0JvOHwowjnaBpSqjPJCWajwzBRDLzV0QGWGKiTZZlE4K7vPIqadWqruEPV5X6bRFHCU7hDC7AhWuowz00oAkEXuANPuDTerXerS/re966ZhUzJ7AAa/ILYBCfWw==</latexit><latexit sha1_base64="/CxJoGcpgoJimdf22nG+xk/jFt0=">AAACGHicbVC7TsMwFL3hWcorwMhiUSExlaRCgrGChbFI9CG1UeS4TmrVToLtIFVRP4OFX2FhACHWbvwNbpuBthzJ0rnn3Gv7niDlTGnH+bHW1jc2t7ZLO+Xdvf2DQ/vouKWSTBLaJAlPZCfAinIW06ZmmtNOKikWAaftYHg39dvPVCqWxI96lFJP4ChmISNYG8m3L3uKRQL7LupF9AkVVa2o+olWC4b07YpTdWZAq8QtSAUKNHx7Ym4hmaCxJhwr1XWdVHs5lpoRTsflXqZoiskQR7RraIwFVV4+W2yMzo3SR2EizYk1mql/J3IslBqJwHQKrAdq2ZuK/3ndTIc3Xs7iNNM0JvOHwowjnaBpSqjPJCWajwzBRDLzV0QGWGKiTZZlE4K7vPIqadWqruEPV5X6bRFHCU7hDC7AhWuowz00oAkEXuANPuDTerXerS/re966ZhUzJ7AAa/ILYBCfWw==</latexit><latexit sha1_base64="/CxJoGcpgoJimdf22nG+xk/jFt0=">AAACGHicbVC7TsMwFL3hWcorwMhiUSExlaRCgrGChbFI9CG1UeS4TmrVToLtIFVRP4OFX2FhACHWbvwNbpuBthzJ0rnn3Gv7niDlTGnH+bHW1jc2t7ZLO+Xdvf2DQ/vouKWSTBLaJAlPZCfAinIW06ZmmtNOKikWAaftYHg39dvPVCqWxI96lFJP4ChmISNYG8m3L3uKRQL7LupF9AkVVa2o+olWC4b07YpTdWZAq8QtSAUKNHx7Ym4hmaCxJhwr1XWdVHs5lpoRTsflXqZoiskQR7RraIwFVV4+W2yMzo3SR2EizYk1mql/J3IslBqJwHQKrAdq2ZuK/3ndTIc3Xs7iNNM0JvOHwowjnaBpSqjPJCWajwzBRDLzV0QGWGKiTZZlE4K7vPIqadWqruEPV5X6bRFHCU7hDC7AhWuowz00oAkEXuANPuDTerXerS/re966ZhUzJ7AAa/ILYBCfWw==</latexit>

U>U = I and V >V = I<latexit sha1_base64="4EUrg+1lvmwxlkQQm0nMTAL+z5o=">AAACD3icbZDNSsNAFIUn9a/Wv6pLN4NFcVUSEXQjFN3oroJpC00tk+mkHTqZhJkbsYS+gRtfxY0LRdy6defbOGmz0NYLAx/n3Mude/xYcA22/W0VFhaXlleKq6W19Y3NrfL2TkNHiaLMpZGIVMsnmgkumQscBGvFipHQF6zpDy8zv3nPlOaRvIVRzDoh6UsecErASN3yoXvnQRRjF5/ja+wBe4AUE9nDY9yYOo3M6ZYrdtWeFJ4HJ4cKyqveLX95vYgmIZNABdG67dgxdFKigFPBxiUv0SwmdEj6rG1QkpDpTjq5Z4wPjNLDQaTMk4An6u+JlIRaj0LfdIYEBnrWy8T/vHYCwVkn5TJOgEk6XRQkAkOEs3BwjytGQYwMEKq4+SumA6IIBRNhyYTgzJ48D43jqmP45qRSu8jjKKI9tI+OkINOUQ1doTpyEUWP6Bm9ojfryXqx3q2PaWvBymd20Z+yPn8AKA6aLw==</latexit><latexit sha1_base64="4EUrg+1lvmwxlkQQm0nMTAL+z5o=">AAACD3icbZDNSsNAFIUn9a/Wv6pLN4NFcVUSEXQjFN3oroJpC00tk+mkHTqZhJkbsYS+gRtfxY0LRdy6defbOGmz0NYLAx/n3Mude/xYcA22/W0VFhaXlleKq6W19Y3NrfL2TkNHiaLMpZGIVMsnmgkumQscBGvFipHQF6zpDy8zv3nPlOaRvIVRzDoh6UsecErASN3yoXvnQRRjF5/ja+wBe4AUE9nDY9yYOo3M6ZYrdtWeFJ4HJ4cKyqveLX95vYgmIZNABdG67dgxdFKigFPBxiUv0SwmdEj6rG1QkpDpTjq5Z4wPjNLDQaTMk4An6u+JlIRaj0LfdIYEBnrWy8T/vHYCwVkn5TJOgEk6XRQkAkOEs3BwjytGQYwMEKq4+SumA6IIBRNhyYTgzJ48D43jqmP45qRSu8jjKKI9tI+OkINOUQ1doTpyEUWP6Bm9ojfryXqx3q2PaWvBymd20Z+yPn8AKA6aLw==</latexit><latexit sha1_base64="4EUrg+1lvmwxlkQQm0nMTAL+z5o=">AAACD3icbZDNSsNAFIUn9a/Wv6pLN4NFcVUSEXQjFN3oroJpC00tk+mkHTqZhJkbsYS+gRtfxY0LRdy6defbOGmz0NYLAx/n3Mude/xYcA22/W0VFhaXlleKq6W19Y3NrfL2TkNHiaLMpZGIVMsnmgkumQscBGvFipHQF6zpDy8zv3nPlOaRvIVRzDoh6UsecErASN3yoXvnQRRjF5/ja+wBe4AUE9nDY9yYOo3M6ZYrdtWeFJ4HJ4cKyqveLX95vYgmIZNABdG67dgxdFKigFPBxiUv0SwmdEj6rG1QkpDpTjq5Z4wPjNLDQaTMk4An6u+JlIRaj0LfdIYEBnrWy8T/vHYCwVkn5TJOgEk6XRQkAkOEs3BwjytGQYwMEKq4+SumA6IIBRNhyYTgzJ48D43jqmP45qRSu8jjKKI9tI+OkINOUQ1doTpyEUWP6Bm9ojfryXqx3q2PaWvBymd20Z+yPn8AKA6aLw==</latexit><latexit sha1_base64="4EUrg+1lvmwxlkQQm0nMTAL+z5o=">AAACD3icbZDNSsNAFIUn9a/Wv6pLN4NFcVUSEXQjFN3oroJpC00tk+mkHTqZhJkbsYS+gRtfxY0LRdy6defbOGmz0NYLAx/n3Mude/xYcA22/W0VFhaXlleKq6W19Y3NrfL2TkNHiaLMpZGIVMsnmgkumQscBGvFipHQF6zpDy8zv3nPlOaRvIVRzDoh6UsecErASN3yoXvnQRRjF5/ja+wBe4AUE9nDY9yYOo3M6ZYrdtWeFJ4HJ4cKyqveLX95vYgmIZNABdG67dgxdFKigFPBxiUv0SwmdEj6rG1QkpDpTjq5Z4wPjNLDQaTMk4An6u+JlIRaj0LfdIYEBnrWy8T/vHYCwVkn5TJOgEk6XRQkAkOEs3BwjytGQYwMEKq4+SumA6IIBRNhyYTgzJ48D43jqmP45qRSu8jjKKI9tI+OkINOUQ1doTpyEUWP6Bm9ojfryXqx3q2PaWvBymd20Z+yPn8AKA6aLw==</latexit>

- Left and right singular vectors are orthogonal:

Page 138: COMP 414/514: Optimization - Algorithms, Complexity and

Matrices- Norms:

..there are more norms to worry about (e.g., operator norms) but we will skip them here..

kAkF =

sX

ij

A2ij

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kAk⇤ =rX

i

�i

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(Frobenius norm) (Nuclear norm)

kAk2 = maxi

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(Spectral norm)

Page 139: COMP 414/514: Optimization - Algorithms, Complexity and

Matrices- Matrix inverses are defined on square matrices

Page 140: COMP 414/514: Optimization - Algorithms, Complexity and

Matrices- Matrix inverses are defined on square matrices

- Matrix inverse definition as a collection of properties:

- is full rank A<latexit sha1_base64="glsIYP92PSMcW8DkNNhZ8pvJRFs=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfRY9eKxBfsBbSib7aRdu9mE3Y1QQn+BFw+KePUnefPfuG1z0NYXFh7emWFn3iARXBvX/XYKa+sbm1vF7dLO7t7+QfnwqKXjVDFssljEqhNQjYJLbBpuBHYShTQKBLaD8d2s3n5CpXksH8wkQT+iQ8lDzqixVuOmX664VXcusgpeDhXIVe+Xv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtShphNrP5otOyZl1BiSMlX3SkLn7eyKjkdaTKLCdETUjvVybmf/VuqkJr/2MyyQ1KNniozAVxMRkdjUZcIXMiIkFyhS3uxI2oooyY7Mp2RC85ZNXoXVR9Sw3Liu12zyOIpzAKZyDB1dQg3uoQxMYIDzDK7w5j86L8+58LFoLTj5zDH/kfP4Akt+MxQ==</latexit><latexit sha1_base64="glsIYP92PSMcW8DkNNhZ8pvJRFs=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfRY9eKxBfsBbSib7aRdu9mE3Y1QQn+BFw+KePUnefPfuG1z0NYXFh7emWFn3iARXBvX/XYKa+sbm1vF7dLO7t7+QfnwqKXjVDFssljEqhNQjYJLbBpuBHYShTQKBLaD8d2s3n5CpXksH8wkQT+iQ8lDzqixVuOmX664VXcusgpeDhXIVe+Xv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtShphNrP5otOyZl1BiSMlX3SkLn7eyKjkdaTKLCdETUjvVybmf/VuqkJr/2MyyQ1KNniozAVxMRkdjUZcIXMiIkFyhS3uxI2oooyY7Mp2RC85ZNXoXVR9Sw3Liu12zyOIpzAKZyDB1dQg3uoQxMYIDzDK7w5j86L8+58LFoLTj5zDH/kfP4Akt+MxQ==</latexit><latexit sha1_base64="glsIYP92PSMcW8DkNNhZ8pvJRFs=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfRY9eKxBfsBbSib7aRdu9mE3Y1QQn+BFw+KePUnefPfuG1z0NYXFh7emWFn3iARXBvX/XYKa+sbm1vF7dLO7t7+QfnwqKXjVDFssljEqhNQjYJLbBpuBHYShTQKBLaD8d2s3n5CpXksH8wkQT+iQ8lDzqixVuOmX664VXcusgpeDhXIVe+Xv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtShphNrP5otOyZl1BiSMlX3SkLn7eyKjkdaTKLCdETUjvVybmf/VuqkJr/2MyyQ1KNniozAVxMRkdjUZcIXMiIkFyhS3uxI2oooyY7Mp2RC85ZNXoXVR9Sw3Liu12zyOIpzAKZyDB1dQg3uoQxMYIDzDK7w5j86L8+58LFoLTj5zDH/kfP4Akt+MxQ==</latexit><latexit sha1_base64="glsIYP92PSMcW8DkNNhZ8pvJRFs=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfRY9eKxBfsBbSib7aRdu9mE3Y1QQn+BFw+KePUnefPfuG1z0NYXFh7emWFn3iARXBvX/XYKa+sbm1vF7dLO7t7+QfnwqKXjVDFssljEqhNQjYJLbBpuBHYShTQKBLaD8d2s3n5CpXksH8wkQT+iQ8lDzqixVuOmX664VXcusgpeDhXIVe+Xv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtShphNrP5otOyZl1BiSMlX3SkLn7eyKjkdaTKLCdETUjvVybmf/VuqkJr/2MyyQ1KNniozAVxMRkdjUZcIXMiIkFyhS3uxI2oooyY7Mp2RC85ZNXoXVR9Sw3Liu12zyOIpzAKZyDB1dQg3uoQxMYIDzDK7w5j86L8+58LFoLTj5zDH/kfP4Akt+MxQ==</latexit>

Page 141: COMP 414/514: Optimization - Algorithms, Complexity and

Matrices- Matrix inverses are defined on square matrices

- Matrix inverse definition as a collection of properties:

- is full rank A<latexit sha1_base64="glsIYP92PSMcW8DkNNhZ8pvJRFs=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfRY9eKxBfsBbSib7aRdu9mE3Y1QQn+BFw+KePUnefPfuG1z0NYXFh7emWFn3iARXBvX/XYKa+sbm1vF7dLO7t7+QfnwqKXjVDFssljEqhNQjYJLbBpuBHYShTQKBLaD8d2s3n5CpXksH8wkQT+iQ8lDzqixVuOmX664VXcusgpeDhXIVe+Xv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtShphNrP5otOyZl1BiSMlX3SkLn7eyKjkdaTKLCdETUjvVybmf/VuqkJr/2MyyQ1KNniozAVxMRkdjUZcIXMiIkFyhS3uxI2oooyY7Mp2RC85ZNXoXVR9Sw3Liu12zyOIpzAKZyDB1dQg3uoQxMYIDzDK7w5j86L8+58LFoLTj5zDH/kfP4Akt+MxQ==</latexit><latexit sha1_base64="glsIYP92PSMcW8DkNNhZ8pvJRFs=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfRY9eKxBfsBbSib7aRdu9mE3Y1QQn+BFw+KePUnefPfuG1z0NYXFh7emWFn3iARXBvX/XYKa+sbm1vF7dLO7t7+QfnwqKXjVDFssljEqhNQjYJLbBpuBHYShTQKBLaD8d2s3n5CpXksH8wkQT+iQ8lDzqixVuOmX664VXcusgpeDhXIVe+Xv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtShphNrP5otOyZl1BiSMlX3SkLn7eyKjkdaTKLCdETUjvVybmf/VuqkJr/2MyyQ1KNniozAVxMRkdjUZcIXMiIkFyhS3uxI2oooyY7Mp2RC85ZNXoXVR9Sw3Liu12zyOIpzAKZyDB1dQg3uoQxMYIDzDK7w5j86L8+58LFoLTj5zDH/kfP4Akt+MxQ==</latexit><latexit sha1_base64="glsIYP92PSMcW8DkNNhZ8pvJRFs=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfRY9eKxBfsBbSib7aRdu9mE3Y1QQn+BFw+KePUnefPfuG1z0NYXFh7emWFn3iARXBvX/XYKa+sbm1vF7dLO7t7+QfnwqKXjVDFssljEqhNQjYJLbBpuBHYShTQKBLaD8d2s3n5CpXksH8wkQT+iQ8lDzqixVuOmX664VXcusgpeDhXIVe+Xv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtShphNrP5otOyZl1BiSMlX3SkLn7eyKjkdaTKLCdETUjvVybmf/VuqkJr/2MyyQ1KNniozAVxMRkdjUZcIXMiIkFyhS3uxI2oooyY7Mp2RC85ZNXoXVR9Sw3Liu12zyOIpzAKZyDB1dQg3uoQxMYIDzDK7w5j86L8+58LFoLTj5zDH/kfP4Akt+MxQ==</latexit><latexit sha1_base64="glsIYP92PSMcW8DkNNhZ8pvJRFs=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfRY9eKxBfsBbSib7aRdu9mE3Y1QQn+BFw+KePUnefPfuG1z0NYXFh7emWFn3iARXBvX/XYKa+sbm1vF7dLO7t7+QfnwqKXjVDFssljEqhNQjYJLbBpuBHYShTQKBLaD8d2s3n5CpXksH8wkQT+iQ8lDzqixVuOmX664VXcusgpeDhXIVe+Xv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtShphNrP5otOyZl1BiSMlX3SkLn7eyKjkdaTKLCdETUjvVybmf/VuqkJr/2MyyQ1KNniozAVxMRkdjUZcIXMiIkFyhS3uxI2oooyY7Mp2RC85ZNXoXVR9Sw3Liu12zyOIpzAKZyDB1dQg3uoQxMYIDzDK7w5j86L8+58LFoLTj5zDH/kfP4Akt+MxQ==</latexit>

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Page 142: COMP 414/514: Optimization - Algorithms, Complexity and

Matrices- Matrix inverses are defined on square matrices

- Matrix inverse definition as a collection of properties:

- is full rank A<latexit sha1_base64="glsIYP92PSMcW8DkNNhZ8pvJRFs=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfRY9eKxBfsBbSib7aRdu9mE3Y1QQn+BFw+KePUnefPfuG1z0NYXFh7emWFn3iARXBvX/XYKa+sbm1vF7dLO7t7+QfnwqKXjVDFssljEqhNQjYJLbBpuBHYShTQKBLaD8d2s3n5CpXksH8wkQT+iQ8lDzqixVuOmX664VXcusgpeDhXIVe+Xv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtShphNrP5otOyZl1BiSMlX3SkLn7eyKjkdaTKLCdETUjvVybmf/VuqkJr/2MyyQ1KNniozAVxMRkdjUZcIXMiIkFyhS3uxI2oooyY7Mp2RC85ZNXoXVR9Sw3Liu12zyOIpzAKZyDB1dQg3uoQxMYIDzDK7w5j86L8+58LFoLTj5zDH/kfP4Akt+MxQ==</latexit><latexit sha1_base64="glsIYP92PSMcW8DkNNhZ8pvJRFs=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfRY9eKxBfsBbSib7aRdu9mE3Y1QQn+BFw+KePUnefPfuG1z0NYXFh7emWFn3iARXBvX/XYKa+sbm1vF7dLO7t7+QfnwqKXjVDFssljEqhNQjYJLbBpuBHYShTQKBLaD8d2s3n5CpXksH8wkQT+iQ8lDzqixVuOmX664VXcusgpeDhXIVe+Xv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtShphNrP5otOyZl1BiSMlX3SkLn7eyKjkdaTKLCdETUjvVybmf/VuqkJr/2MyyQ1KNniozAVxMRkdjUZcIXMiIkFyhS3uxI2oooyY7Mp2RC85ZNXoXVR9Sw3Liu12zyOIpzAKZyDB1dQg3uoQxMYIDzDK7w5j86L8+58LFoLTj5zDH/kfP4Akt+MxQ==</latexit><latexit sha1_base64="glsIYP92PSMcW8DkNNhZ8pvJRFs=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfRY9eKxBfsBbSib7aRdu9mE3Y1QQn+BFw+KePUnefPfuG1z0NYXFh7emWFn3iARXBvX/XYKa+sbm1vF7dLO7t7+QfnwqKXjVDFssljEqhNQjYJLbBpuBHYShTQKBLaD8d2s3n5CpXksH8wkQT+iQ8lDzqixVuOmX664VXcusgpeDhXIVe+Xv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtShphNrP5otOyZl1BiSMlX3SkLn7eyKjkdaTKLCdETUjvVybmf/VuqkJr/2MyyQ1KNniozAVxMRkdjUZcIXMiIkFyhS3uxI2oooyY7Mp2RC85ZNXoXVR9Sw3Liu12zyOIpzAKZyDB1dQg3uoQxMYIDzDK7w5j86L8+58LFoLTj5zDH/kfP4Akt+MxQ==</latexit><latexit sha1_base64="glsIYP92PSMcW8DkNNhZ8pvJRFs=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfRY9eKxBfsBbSib7aRdu9mE3Y1QQn+BFw+KePUnefPfuG1z0NYXFh7emWFn3iARXBvX/XYKa+sbm1vF7dLO7t7+QfnwqKXjVDFssljEqhNQjYJLbBpuBHYShTQKBLaD8d2s3n5CpXksH8wkQT+iQ8lDzqixVuOmX664VXcusgpeDhXIVe+Xv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtShphNrP5otOyZl1BiSMlX3SkLn7eyKjkdaTKLCdETUjvVybmf/VuqkJr/2MyyQ1KNniozAVxMRkdjUZcIXMiIkFyhS3uxI2oooyY7Mp2RC85ZNXoXVR9Sw3Liu12zyOIpzAKZyDB1dQg3uoQxMYIDzDK7w5j86L8+58LFoLTj5zDH/kfP4Akt+MxQ==</latexit>

- has empty nullspaceA<latexit sha1_base64="glsIYP92PSMcW8DkNNhZ8pvJRFs=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfRY9eKxBfsBbSib7aRdu9mE3Y1QQn+BFw+KePUnefPfuG1z0NYXFh7emWFn3iARXBvX/XYKa+sbm1vF7dLO7t7+QfnwqKXjVDFssljEqhNQjYJLbBpuBHYShTQKBLaD8d2s3n5CpXksH8wkQT+iQ8lDzqixVuOmX664VXcusgpeDhXIVe+Xv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtShphNrP5otOyZl1BiSMlX3SkLn7eyKjkdaTKLCdETUjvVybmf/VuqkJr/2MyyQ1KNniozAVxMRkdjUZcIXMiIkFyhS3uxI2oooyY7Mp2RC85ZNXoXVR9Sw3Liu12zyOIpzAKZyDB1dQg3uoQxMYIDzDK7w5j86L8+58LFoLTj5zDH/kfP4Akt+MxQ==</latexit><latexit sha1_base64="glsIYP92PSMcW8DkNNhZ8pvJRFs=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfRY9eKxBfsBbSib7aRdu9mE3Y1QQn+BFw+KePUnefPfuG1z0NYXFh7emWFn3iARXBvX/XYKa+sbm1vF7dLO7t7+QfnwqKXjVDFssljEqhNQjYJLbBpuBHYShTQKBLaD8d2s3n5CpXksH8wkQT+iQ8lDzqixVuOmX664VXcusgpeDhXIVe+Xv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtShphNrP5otOyZl1BiSMlX3SkLn7eyKjkdaTKLCdETUjvVybmf/VuqkJr/2MyyQ1KNniozAVxMRkdjUZcIXMiIkFyhS3uxI2oooyY7Mp2RC85ZNXoXVR9Sw3Liu12zyOIpzAKZyDB1dQg3uoQxMYIDzDK7w5j86L8+58LFoLTj5zDH/kfP4Akt+MxQ==</latexit><latexit sha1_base64="glsIYP92PSMcW8DkNNhZ8pvJRFs=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfRY9eKxBfsBbSib7aRdu9mE3Y1QQn+BFw+KePUnefPfuG1z0NYXFh7emWFn3iARXBvX/XYKa+sbm1vF7dLO7t7+QfnwqKXjVDFssljEqhNQjYJLbBpuBHYShTQKBLaD8d2s3n5CpXksH8wkQT+iQ8lDzqixVuOmX664VXcusgpeDhXIVe+Xv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtShphNrP5otOyZl1BiSMlX3SkLn7eyKjkdaTKLCdETUjvVybmf/VuqkJr/2MyyQ1KNniozAVxMRkdjUZcIXMiIkFyhS3uxI2oooyY7Mp2RC85ZNXoXVR9Sw3Liu12zyOIpzAKZyDB1dQg3uoQxMYIDzDK7w5j86L8+58LFoLTj5zDH/kfP4Akt+MxQ==</latexit><latexit sha1_base64="glsIYP92PSMcW8DkNNhZ8pvJRFs=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfRY9eKxBfsBbSib7aRdu9mE3Y1QQn+BFw+KePUnefPfuG1z0NYXFh7emWFn3iARXBvX/XYKa+sbm1vF7dLO7t7+QfnwqKXjVDFssljEqhNQjYJLbBpuBHYShTQKBLaD8d2s3n5CpXksH8wkQT+iQ8lDzqixVuOmX664VXcusgpeDhXIVe+Xv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtShphNrP5otOyZl1BiSMlX3SkLn7eyKjkdaTKLCdETUjvVybmf/VuqkJr/2MyyQ1KNniozAVxMRkdjUZcIXMiIkFyhS3uxI2oooyY7Mp2RC85ZNXoXVR9Sw3Liu12zyOIpzAKZyDB1dQg3uoQxMYIDzDK7w5j86L8+58LFoLTj5zDH/kfP4Akt+MxQ==</latexit>

- The equation has only the trivial solution Ax = 0<latexit sha1_base64="yY1xCWv4apZjphe88GtRt86JNMg=">AAAB7XicbZDLSgMxFIZPvNZ6q7p0EyyCqzIjgm6EqhuXFewF2qFk0kwbm0mGJCOWoe/gxoUibn0fd76NaTsLbf0h8PGfc8g5f5gIbqznfaOl5ZXVtfXCRnFza3tnt7S33zAq1ZTVqRJKt0JimOCS1S23grUSzUgcCtYMhzeTevORacOVvLejhAUx6UsecUqssxpXT/gSe91S2at4U+FF8HMoQ65at/TV6SmaxkxaKogxbd9LbJARbTkVbFzspIYlhA5Jn7UdShIzE2TTbcf42Dk9HCntnrR46v6eyEhszCgOXWdM7MDM1ybmf7V2aqOLIOMySS2TdPZRlApsFZ6cjntcM2rFyAGhmrtdMR0QTah1ARVdCP78yYvQOK34ju/OytXrPI4CHMIRnIAP51CFW6hBHSg8wDO8whtS6AW9o49Z6xLKZw7gj9DnDwvHjhw=</latexit><latexit sha1_base64="yY1xCWv4apZjphe88GtRt86JNMg=">AAAB7XicbZDLSgMxFIZPvNZ6q7p0EyyCqzIjgm6EqhuXFewF2qFk0kwbm0mGJCOWoe/gxoUibn0fd76NaTsLbf0h8PGfc8g5f5gIbqznfaOl5ZXVtfXCRnFza3tnt7S33zAq1ZTVqRJKt0JimOCS1S23grUSzUgcCtYMhzeTevORacOVvLejhAUx6UsecUqssxpXT/gSe91S2at4U+FF8HMoQ65at/TV6SmaxkxaKogxbd9LbJARbTkVbFzspIYlhA5Jn7UdShIzE2TTbcf42Dk9HCntnrR46v6eyEhszCgOXWdM7MDM1ybmf7V2aqOLIOMySS2TdPZRlApsFZ6cjntcM2rFyAGhmrtdMR0QTah1ARVdCP78yYvQOK34ju/OytXrPI4CHMIRnIAP51CFW6hBHSg8wDO8whtS6AW9o49Z6xLKZw7gj9DnDwvHjhw=</latexit><latexit sha1_base64="yY1xCWv4apZjphe88GtRt86JNMg=">AAAB7XicbZDLSgMxFIZPvNZ6q7p0EyyCqzIjgm6EqhuXFewF2qFk0kwbm0mGJCOWoe/gxoUibn0fd76NaTsLbf0h8PGfc8g5f5gIbqznfaOl5ZXVtfXCRnFza3tnt7S33zAq1ZTVqRJKt0JimOCS1S23grUSzUgcCtYMhzeTevORacOVvLejhAUx6UsecUqssxpXT/gSe91S2at4U+FF8HMoQ65at/TV6SmaxkxaKogxbd9LbJARbTkVbFzspIYlhA5Jn7UdShIzE2TTbcf42Dk9HCntnrR46v6eyEhszCgOXWdM7MDM1ybmf7V2aqOLIOMySS2TdPZRlApsFZ6cjntcM2rFyAGhmrtdMR0QTah1ARVdCP78yYvQOK34ju/OytXrPI4CHMIRnIAP51CFW6hBHSg8wDO8whtS6AW9o49Z6xLKZw7gj9DnDwvHjhw=</latexit><latexit sha1_base64="yY1xCWv4apZjphe88GtRt86JNMg=">AAAB7XicbZDLSgMxFIZPvNZ6q7p0EyyCqzIjgm6EqhuXFewF2qFk0kwbm0mGJCOWoe/gxoUibn0fd76NaTsLbf0h8PGfc8g5f5gIbqznfaOl5ZXVtfXCRnFza3tnt7S33zAq1ZTVqRJKt0JimOCS1S23grUSzUgcCtYMhzeTevORacOVvLejhAUx6UsecUqssxpXT/gSe91S2at4U+FF8HMoQ65at/TV6SmaxkxaKogxbd9LbJARbTkVbFzspIYlhA5Jn7UdShIzE2TTbcf42Dk9HCntnrR46v6eyEhszCgOXWdM7MDM1ybmf7V2aqOLIOMySS2TdPZRlApsFZ6cjntcM2rFyAGhmrtdMR0QTah1ARVdCP78yYvQOK34ju/OytXrPI4CHMIRnIAP51CFW6hBHSg8wDO8whtS6AW9o49Z6xLKZw7gj9DnDwvHjhw=</latexit>

x = 0<latexit sha1_base64="TMIHVM7cjPNIpSJEQSU+/pH7FkY=">AAAB7HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfQiFL14rGBqoQ1ls520SzebsLsRS+hv8OJBEa/+IG/+G7dtDtr6wsLDOzPszBumgmvjut9OaWV1bX2jvFnZ2t7Z3avuH7R0kimGPktEotoh1Si4RN9wI7CdKqRxKPAhHN1M6w+PqDRP5L0ZpxjEdCB5xBk11vKfyBVxe9WaW3dnIsvgFVCDQs1e9avbT1gWozRMUK07npuaIKfKcCZwUulmGlPKRnSAHYuSxqiDfLbshJxYp0+iRNknDZm5vydyGms9jkPbGVMz1Iu1qflfrZOZ6DLIuUwzg5LNP4oyQUxCppeTPlfIjBhboExxuythQ6ooMzafig3BWzx5GVpndc/y3XmtcV3EUYYjOIZT8OACGnALTfCBAYdneIU3RzovzrvzMW8tOcXMIfyR8/kDgwON0Q==</latexit><latexit sha1_base64="TMIHVM7cjPNIpSJEQSU+/pH7FkY=">AAAB7HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfQiFL14rGBqoQ1ls520SzebsLsRS+hv8OJBEa/+IG/+G7dtDtr6wsLDOzPszBumgmvjut9OaWV1bX2jvFnZ2t7Z3avuH7R0kimGPktEotoh1Si4RN9wI7CdKqRxKPAhHN1M6w+PqDRP5L0ZpxjEdCB5xBk11vKfyBVxe9WaW3dnIsvgFVCDQs1e9avbT1gWozRMUK07npuaIKfKcCZwUulmGlPKRnSAHYuSxqiDfLbshJxYp0+iRNknDZm5vydyGms9jkPbGVMz1Iu1qflfrZOZ6DLIuUwzg5LNP4oyQUxCppeTPlfIjBhboExxuythQ6ooMzafig3BWzx5GVpndc/y3XmtcV3EUYYjOIZT8OACGnALTfCBAYdneIU3RzovzrvzMW8tOcXMIfyR8/kDgwON0Q==</latexit><latexit sha1_base64="TMIHVM7cjPNIpSJEQSU+/pH7FkY=">AAAB7HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfQiFL14rGBqoQ1ls520SzebsLsRS+hv8OJBEa/+IG/+G7dtDtr6wsLDOzPszBumgmvjut9OaWV1bX2jvFnZ2t7Z3avuH7R0kimGPktEotoh1Si4RN9wI7CdKqRxKPAhHN1M6w+PqDRP5L0ZpxjEdCB5xBk11vKfyBVxe9WaW3dnIsvgFVCDQs1e9avbT1gWozRMUK07npuaIKfKcCZwUulmGlPKRnSAHYuSxqiDfLbshJxYp0+iRNknDZm5vydyGms9jkPbGVMz1Iu1qflfrZOZ6DLIuUwzg5LNP4oyQUxCppeTPlfIjBhboExxuythQ6ooMzafig3BWzx5GVpndc/y3XmtcV3EUYYjOIZT8OACGnALTfCBAYdneIU3RzovzrvzMW8tOcXMIfyR8/kDgwON0Q==</latexit><latexit sha1_base64="TMIHVM7cjPNIpSJEQSU+/pH7FkY=">AAAB7HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfQiFL14rGBqoQ1ls520SzebsLsRS+hv8OJBEa/+IG/+G7dtDtr6wsLDOzPszBumgmvjut9OaWV1bX2jvFnZ2t7Z3avuH7R0kimGPktEotoh1Si4RN9wI7CdKqRxKPAhHN1M6w+PqDRP5L0ZpxjEdCB5xBk11vKfyBVxe9WaW3dnIsvgFVCDQs1e9avbT1gWozRMUK07npuaIKfKcCZwUulmGlPKRnSAHYuSxqiDfLbshJxYp0+iRNknDZm5vydyGms9jkPbGVMz1Iu1qflfrZOZ6DLIuUwzg5LNP4oyQUxCppeTPlfIjBhboExxuythQ6ooMzafig3BWzx5GVpndc/y3XmtcV3EUYYjOIZT8OACGnALTfCBAYdneIU3RzovzrvzMW8tOcXMIfyR8/kDgwON0Q==</latexit>

Page 143: COMP 414/514: Optimization - Algorithms, Complexity and

Matrices- Matrix inverses are defined on square matrices

- Matrix inverse definition as a collection of properties:

- is full rank A<latexit sha1_base64="glsIYP92PSMcW8DkNNhZ8pvJRFs=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfRY9eKxBfsBbSib7aRdu9mE3Y1QQn+BFw+KePUnefPfuG1z0NYXFh7emWFn3iARXBvX/XYKa+sbm1vF7dLO7t7+QfnwqKXjVDFssljEqhNQjYJLbBpuBHYShTQKBLaD8d2s3n5CpXksH8wkQT+iQ8lDzqixVuOmX664VXcusgpeDhXIVe+Xv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtShphNrP5otOyZl1BiSMlX3SkLn7eyKjkdaTKLCdETUjvVybmf/VuqkJr/2MyyQ1KNniozAVxMRkdjUZcIXMiIkFyhS3uxI2oooyY7Mp2RC85ZNXoXVR9Sw3Liu12zyOIpzAKZyDB1dQg3uoQxMYIDzDK7w5j86L8+58LFoLTj5zDH/kfP4Akt+MxQ==</latexit><latexit sha1_base64="glsIYP92PSMcW8DkNNhZ8pvJRFs=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfRY9eKxBfsBbSib7aRdu9mE3Y1QQn+BFw+KePUnefPfuG1z0NYXFh7emWFn3iARXBvX/XYKa+sbm1vF7dLO7t7+QfnwqKXjVDFssljEqhNQjYJLbBpuBHYShTQKBLaD8d2s3n5CpXksH8wkQT+iQ8lDzqixVuOmX664VXcusgpeDhXIVe+Xv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtShphNrP5otOyZl1BiSMlX3SkLn7eyKjkdaTKLCdETUjvVybmf/VuqkJr/2MyyQ1KNniozAVxMRkdjUZcIXMiIkFyhS3uxI2oooyY7Mp2RC85ZNXoXVR9Sw3Liu12zyOIpzAKZyDB1dQg3uoQxMYIDzDK7w5j86L8+58LFoLTj5zDH/kfP4Akt+MxQ==</latexit><latexit sha1_base64="glsIYP92PSMcW8DkNNhZ8pvJRFs=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfRY9eKxBfsBbSib7aRdu9mE3Y1QQn+BFw+KePUnefPfuG1z0NYXFh7emWFn3iARXBvX/XYKa+sbm1vF7dLO7t7+QfnwqKXjVDFssljEqhNQjYJLbBpuBHYShTQKBLaD8d2s3n5CpXksH8wkQT+iQ8lDzqixVuOmX664VXcusgpeDhXIVe+Xv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtShphNrP5otOyZl1BiSMlX3SkLn7eyKjkdaTKLCdETUjvVybmf/VuqkJr/2MyyQ1KNniozAVxMRkdjUZcIXMiIkFyhS3uxI2oooyY7Mp2RC85ZNXoXVR9Sw3Liu12zyOIpzAKZyDB1dQg3uoQxMYIDzDK7w5j86L8+58LFoLTj5zDH/kfP4Akt+MxQ==</latexit><latexit sha1_base64="glsIYP92PSMcW8DkNNhZ8pvJRFs=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfRY9eKxBfsBbSib7aRdu9mE3Y1QQn+BFw+KePUnefPfuG1z0NYXFh7emWFn3iARXBvX/XYKa+sbm1vF7dLO7t7+QfnwqKXjVDFssljEqhNQjYJLbBpuBHYShTQKBLaD8d2s3n5CpXksH8wkQT+iQ8lDzqixVuOmX664VXcusgpeDhXIVe+Xv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtShphNrP5otOyZl1BiSMlX3SkLn7eyKjkdaTKLCdETUjvVybmf/VuqkJr/2MyyQ1KNniozAVxMRkdjUZcIXMiIkFyhS3uxI2oooyY7Mp2RC85ZNXoXVR9Sw3Liu12zyOIpzAKZyDB1dQg3uoQxMYIDzDK7w5j86L8+58LFoLTj5zDH/kfP4Akt+MxQ==</latexit>

- has empty nullspaceA<latexit sha1_base64="glsIYP92PSMcW8DkNNhZ8pvJRFs=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfRY9eKxBfsBbSib7aRdu9mE3Y1QQn+BFw+KePUnefPfuG1z0NYXFh7emWFn3iARXBvX/XYKa+sbm1vF7dLO7t7+QfnwqKXjVDFssljEqhNQjYJLbBpuBHYShTQKBLaD8d2s3n5CpXksH8wkQT+iQ8lDzqixVuOmX664VXcusgpeDhXIVe+Xv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtShphNrP5otOyZl1BiSMlX3SkLn7eyKjkdaTKLCdETUjvVybmf/VuqkJr/2MyyQ1KNniozAVxMRkdjUZcIXMiIkFyhS3uxI2oooyY7Mp2RC85ZNXoXVR9Sw3Liu12zyOIpzAKZyDB1dQg3uoQxMYIDzDK7w5j86L8+58LFoLTj5zDH/kfP4Akt+MxQ==</latexit><latexit sha1_base64="glsIYP92PSMcW8DkNNhZ8pvJRFs=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfRY9eKxBfsBbSib7aRdu9mE3Y1QQn+BFw+KePUnefPfuG1z0NYXFh7emWFn3iARXBvX/XYKa+sbm1vF7dLO7t7+QfnwqKXjVDFssljEqhNQjYJLbBpuBHYShTQKBLaD8d2s3n5CpXksH8wkQT+iQ8lDzqixVuOmX664VXcusgpeDhXIVe+Xv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtShphNrP5otOyZl1BiSMlX3SkLn7eyKjkdaTKLCdETUjvVybmf/VuqkJr/2MyyQ1KNniozAVxMRkdjUZcIXMiIkFyhS3uxI2oooyY7Mp2RC85ZNXoXVR9Sw3Liu12zyOIpzAKZyDB1dQg3uoQxMYIDzDK7w5j86L8+58LFoLTj5zDH/kfP4Akt+MxQ==</latexit><latexit sha1_base64="glsIYP92PSMcW8DkNNhZ8pvJRFs=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfRY9eKxBfsBbSib7aRdu9mE3Y1QQn+BFw+KePUnefPfuG1z0NYXFh7emWFn3iARXBvX/XYKa+sbm1vF7dLO7t7+QfnwqKXjVDFssljEqhNQjYJLbBpuBHYShTQKBLaD8d2s3n5CpXksH8wkQT+iQ8lDzqixVuOmX664VXcusgpeDhXIVe+Xv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtShphNrP5otOyZl1BiSMlX3SkLn7eyKjkdaTKLCdETUjvVybmf/VuqkJr/2MyyQ1KNniozAVxMRkdjUZcIXMiIkFyhS3uxI2oooyY7Mp2RC85ZNXoXVR9Sw3Liu12zyOIpzAKZyDB1dQg3uoQxMYIDzDK7w5j86L8+58LFoLTj5zDH/kfP4Akt+MxQ==</latexit><latexit sha1_base64="glsIYP92PSMcW8DkNNhZ8pvJRFs=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfRY9eKxBfsBbSib7aRdu9mE3Y1QQn+BFw+KePUnefPfuG1z0NYXFh7emWFn3iARXBvX/XYKa+sbm1vF7dLO7t7+QfnwqKXjVDFssljEqhNQjYJLbBpuBHYShTQKBLaD8d2s3n5CpXksH8wkQT+iQ8lDzqixVuOmX664VXcusgpeDhXIVe+Xv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtShphNrP5otOyZl1BiSMlX3SkLn7eyKjkdaTKLCdETUjvVybmf/VuqkJr/2MyyQ1KNniozAVxMRkdjUZcIXMiIkFyhS3uxI2oooyY7Mp2RC85ZNXoXVR9Sw3Liu12zyOIpzAKZyDB1dQg3uoQxMYIDzDK7w5j86L8+58LFoLTj5zDH/kfP4Akt+MxQ==</latexit>

- The equation has only the trivial solution Ax = 0<latexit sha1_base64="yY1xCWv4apZjphe88GtRt86JNMg=">AAAB7XicbZDLSgMxFIZPvNZ6q7p0EyyCqzIjgm6EqhuXFewF2qFk0kwbm0mGJCOWoe/gxoUibn0fd76NaTsLbf0h8PGfc8g5f5gIbqznfaOl5ZXVtfXCRnFza3tnt7S33zAq1ZTVqRJKt0JimOCS1S23grUSzUgcCtYMhzeTevORacOVvLejhAUx6UsecUqssxpXT/gSe91S2at4U+FF8HMoQ65at/TV6SmaxkxaKogxbd9LbJARbTkVbFzspIYlhA5Jn7UdShIzE2TTbcf42Dk9HCntnrR46v6eyEhszCgOXWdM7MDM1ybmf7V2aqOLIOMySS2TdPZRlApsFZ6cjntcM2rFyAGhmrtdMR0QTah1ARVdCP78yYvQOK34ju/OytXrPI4CHMIRnIAP51CFW6hBHSg8wDO8whtS6AW9o49Z6xLKZw7gj9DnDwvHjhw=</latexit><latexit sha1_base64="yY1xCWv4apZjphe88GtRt86JNMg=">AAAB7XicbZDLSgMxFIZPvNZ6q7p0EyyCqzIjgm6EqhuXFewF2qFk0kwbm0mGJCOWoe/gxoUibn0fd76NaTsLbf0h8PGfc8g5f5gIbqznfaOl5ZXVtfXCRnFza3tnt7S33zAq1ZTVqRJKt0JimOCS1S23grUSzUgcCtYMhzeTevORacOVvLejhAUx6UsecUqssxpXT/gSe91S2at4U+FF8HMoQ65at/TV6SmaxkxaKogxbd9LbJARbTkVbFzspIYlhA5Jn7UdShIzE2TTbcf42Dk9HCntnrR46v6eyEhszCgOXWdM7MDM1ybmf7V2aqOLIOMySS2TdPZRlApsFZ6cjntcM2rFyAGhmrtdMR0QTah1ARVdCP78yYvQOK34ju/OytXrPI4CHMIRnIAP51CFW6hBHSg8wDO8whtS6AW9o49Z6xLKZw7gj9DnDwvHjhw=</latexit><latexit sha1_base64="yY1xCWv4apZjphe88GtRt86JNMg=">AAAB7XicbZDLSgMxFIZPvNZ6q7p0EyyCqzIjgm6EqhuXFewF2qFk0kwbm0mGJCOWoe/gxoUibn0fd76NaTsLbf0h8PGfc8g5f5gIbqznfaOl5ZXVtfXCRnFza3tnt7S33zAq1ZTVqRJKt0JimOCS1S23grUSzUgcCtYMhzeTevORacOVvLejhAUx6UsecUqssxpXT/gSe91S2at4U+FF8HMoQ65at/TV6SmaxkxaKogxbd9LbJARbTkVbFzspIYlhA5Jn7UdShIzE2TTbcf42Dk9HCntnrR46v6eyEhszCgOXWdM7MDM1ybmf7V2aqOLIOMySS2TdPZRlApsFZ6cjntcM2rFyAGhmrtdMR0QTah1ARVdCP78yYvQOK34ju/OytXrPI4CHMIRnIAP51CFW6hBHSg8wDO8whtS6AW9o49Z6xLKZw7gj9DnDwvHjhw=</latexit><latexit sha1_base64="yY1xCWv4apZjphe88GtRt86JNMg=">AAAB7XicbZDLSgMxFIZPvNZ6q7p0EyyCqzIjgm6EqhuXFewF2qFk0kwbm0mGJCOWoe/gxoUibn0fd76NaTsLbf0h8PGfc8g5f5gIbqznfaOl5ZXVtfXCRnFza3tnt7S33zAq1ZTVqRJKt0JimOCS1S23grUSzUgcCtYMhzeTevORacOVvLejhAUx6UsecUqssxpXT/gSe91S2at4U+FF8HMoQ65at/TV6SmaxkxaKogxbd9LbJARbTkVbFzspIYlhA5Jn7UdShIzE2TTbcf42Dk9HCntnrR46v6eyEhszCgOXWdM7MDM1ybmf7V2aqOLIOMySS2TdPZRlApsFZ6cjntcM2rFyAGhmrtdMR0QTah1ARVdCP78yYvQOK34ju/OytXrPI4CHMIRnIAP51CFW6hBHSg8wDO8whtS6AW9o49Z6xLKZw7gj9DnDwvHjhw=</latexit>

x = 0<latexit sha1_base64="TMIHVM7cjPNIpSJEQSU+/pH7FkY=">AAAB7HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfQiFL14rGBqoQ1ls520SzebsLsRS+hv8OJBEa/+IG/+G7dtDtr6wsLDOzPszBumgmvjut9OaWV1bX2jvFnZ2t7Z3avuH7R0kimGPktEotoh1Si4RN9wI7CdKqRxKPAhHN1M6w+PqDRP5L0ZpxjEdCB5xBk11vKfyBVxe9WaW3dnIsvgFVCDQs1e9avbT1gWozRMUK07npuaIKfKcCZwUulmGlPKRnSAHYuSxqiDfLbshJxYp0+iRNknDZm5vydyGms9jkPbGVMz1Iu1qflfrZOZ6DLIuUwzg5LNP4oyQUxCppeTPlfIjBhboExxuythQ6ooMzafig3BWzx5GVpndc/y3XmtcV3EUYYjOIZT8OACGnALTfCBAYdneIU3RzovzrvzMW8tOcXMIfyR8/kDgwON0Q==</latexit><latexit sha1_base64="TMIHVM7cjPNIpSJEQSU+/pH7FkY=">AAAB7HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfQiFL14rGBqoQ1ls520SzebsLsRS+hv8OJBEa/+IG/+G7dtDtr6wsLDOzPszBumgmvjut9OaWV1bX2jvFnZ2t7Z3avuH7R0kimGPktEotoh1Si4RN9wI7CdKqRxKPAhHN1M6w+PqDRP5L0ZpxjEdCB5xBk11vKfyBVxe9WaW3dnIsvgFVCDQs1e9avbT1gWozRMUK07npuaIKfKcCZwUulmGlPKRnSAHYuSxqiDfLbshJxYp0+iRNknDZm5vydyGms9jkPbGVMz1Iu1qflfrZOZ6DLIuUwzg5LNP4oyQUxCppeTPlfIjBhboExxuythQ6ooMzafig3BWzx5GVpndc/y3XmtcV3EUYYjOIZT8OACGnALTfCBAYdneIU3RzovzrvzMW8tOcXMIfyR8/kDgwON0Q==</latexit><latexit sha1_base64="TMIHVM7cjPNIpSJEQSU+/pH7FkY=">AAAB7HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfQiFL14rGBqoQ1ls520SzebsLsRS+hv8OJBEa/+IG/+G7dtDtr6wsLDOzPszBumgmvjut9OaWV1bX2jvFnZ2t7Z3avuH7R0kimGPktEotoh1Si4RN9wI7CdKqRxKPAhHN1M6w+PqDRP5L0ZpxjEdCB5xBk11vKfyBVxe9WaW3dnIsvgFVCDQs1e9avbT1gWozRMUK07npuaIKfKcCZwUulmGlPKRnSAHYuSxqiDfLbshJxYp0+iRNknDZm5vydyGms9jkPbGVMz1Iu1qflfrZOZ6DLIuUwzg5LNP4oyQUxCppeTPlfIjBhboExxuythQ6ooMzafig3BWzx5GVpndc/y3XmtcV3EUYYjOIZT8OACGnALTfCBAYdneIU3RzovzrvzMW8tOcXMIfyR8/kDgwON0Q==</latexit><latexit sha1_base64="TMIHVM7cjPNIpSJEQSU+/pH7FkY=">AAAB7HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfQiFL14rGBqoQ1ls520SzebsLsRS+hv8OJBEa/+IG/+G7dtDtr6wsLDOzPszBumgmvjut9OaWV1bX2jvFnZ2t7Z3avuH7R0kimGPktEotoh1Si4RN9wI7CdKqRxKPAhHN1M6w+PqDRP5L0ZpxjEdCB5xBk11vKfyBVxe9WaW3dnIsvgFVCDQs1e9avbT1gWozRMUK07npuaIKfKcCZwUulmGlPKRnSAHYuSxqiDfLbshJxYp0+iRNknDZm5vydyGms9jkPbGVMz1Iu1qflfrZOZ6DLIuUwzg5LNP4oyQUxCppeTPlfIjBhboExxuythQ6ooMzafig3BWzx5GVpndc/y3XmtcV3EUYYjOIZT8OACGnALTfCBAYdneIU3RzovzrvzMW8tOcXMIfyR8/kDgwON0Q==</latexit>

- The linear system has a unique solutionAx = b<latexit sha1_base64="8Viwh2D00yJ5mcgJngUqqpm71Gs=">AAAB7XicbZDLSgMxFIZPvNZ6q7p0EyyCqzIjgm6EqhuXFewF2qFk0kwbm0mGJCOWoe/gxoUibn0fd76NaTsLbf0h8PGfc8g5f5gIbqznfaOl5ZXVtfXCRnFza3tnt7S33zAq1ZTVqRJKt0JimOCS1S23grUSzUgcCtYMhzeTevORacOVvLejhAUx6UsecUqssxpXT/gSh91S2at4U+FF8HMoQ65at/TV6SmaxkxaKogxbd9LbJARbTkVbFzspIYlhA5Jn7UdShIzE2TTbcf42Dk9HCntnrR46v6eyEhszCgOXWdM7MDM1ybmf7V2aqOLIOMySS2TdPZRlApsFZ6cjntcM2rFyAGhmrtdMR0QTah1ARVdCP78yYvQOK34ju/OytXrPI4CHMIRnIAP51CFW6hBHSg8wDO8whtS6AW9o49Z6xLKZw7gj9DnD1ePjk4=</latexit><latexit sha1_base64="8Viwh2D00yJ5mcgJngUqqpm71Gs=">AAAB7XicbZDLSgMxFIZPvNZ6q7p0EyyCqzIjgm6EqhuXFewF2qFk0kwbm0mGJCOWoe/gxoUibn0fd76NaTsLbf0h8PGfc8g5f5gIbqznfaOl5ZXVtfXCRnFza3tnt7S33zAq1ZTVqRJKt0JimOCS1S23grUSzUgcCtYMhzeTevORacOVvLejhAUx6UsecUqssxpXT/gSh91S2at4U+FF8HMoQ65at/TV6SmaxkxaKogxbd9LbJARbTkVbFzspIYlhA5Jn7UdShIzE2TTbcf42Dk9HCntnrR46v6eyEhszCgOXWdM7MDM1ybmf7V2aqOLIOMySS2TdPZRlApsFZ6cjntcM2rFyAGhmrtdMR0QTah1ARVdCP78yYvQOK34ju/OytXrPI4CHMIRnIAP51CFW6hBHSg8wDO8whtS6AW9o49Z6xLKZw7gj9DnD1ePjk4=</latexit><latexit sha1_base64="8Viwh2D00yJ5mcgJngUqqpm71Gs=">AAAB7XicbZDLSgMxFIZPvNZ6q7p0EyyCqzIjgm6EqhuXFewF2qFk0kwbm0mGJCOWoe/gxoUibn0fd76NaTsLbf0h8PGfc8g5f5gIbqznfaOl5ZXVtfXCRnFza3tnt7S33zAq1ZTVqRJKt0JimOCS1S23grUSzUgcCtYMhzeTevORacOVvLejhAUx6UsecUqssxpXT/gSh91S2at4U+FF8HMoQ65at/TV6SmaxkxaKogxbd9LbJARbTkVbFzspIYlhA5Jn7UdShIzE2TTbcf42Dk9HCntnrR46v6eyEhszCgOXWdM7MDM1ybmf7V2aqOLIOMySS2TdPZRlApsFZ6cjntcM2rFyAGhmrtdMR0QTah1ARVdCP78yYvQOK34ju/OytXrPI4CHMIRnIAP51CFW6hBHSg8wDO8whtS6AW9o49Z6xLKZw7gj9DnD1ePjk4=</latexit><latexit sha1_base64="8Viwh2D00yJ5mcgJngUqqpm71Gs=">AAAB7XicbZDLSgMxFIZPvNZ6q7p0EyyCqzIjgm6EqhuXFewF2qFk0kwbm0mGJCOWoe/gxoUibn0fd76NaTsLbf0h8PGfc8g5f5gIbqznfaOl5ZXVtfXCRnFza3tnt7S33zAq1ZTVqRJKt0JimOCS1S23grUSzUgcCtYMhzeTevORacOVvLejhAUx6UsecUqssxpXT/gSh91S2at4U+FF8HMoQ65at/TV6SmaxkxaKogxbd9LbJARbTkVbFzspIYlhA5Jn7UdShIzE2TTbcf42Dk9HCntnrR46v6eyEhszCgOXWdM7MDM1ybmf7V2aqOLIOMySS2TdPZRlApsFZ6cjntcM2rFyAGhmrtdMR0QTah1ARVdCP78yYvQOK34ju/OytXrPI4CHMIRnIAP51CFW6hBHSg8wDO8whtS6AW9o49Z6xLKZw7gj9DnD1ePjk4=</latexit>

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- The columns and rows of are linearly independentA<latexit sha1_base64="glsIYP92PSMcW8DkNNhZ8pvJRFs=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfRY9eKxBfsBbSib7aRdu9mE3Y1QQn+BFw+KePUnefPfuG1z0NYXFh7emWFn3iARXBvX/XYKa+sbm1vF7dLO7t7+QfnwqKXjVDFssljEqhNQjYJLbBpuBHYShTQKBLaD8d2s3n5CpXksH8wkQT+iQ8lDzqixVuOmX664VXcusgpeDhXIVe+Xv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtShphNrP5otOyZl1BiSMlX3SkLn7eyKjkdaTKLCdETUjvVybmf/VuqkJr/2MyyQ1KNniozAVxMRkdjUZcIXMiIkFyhS3uxI2oooyY7Mp2RC85ZNXoXVR9Sw3Liu12zyOIpzAKZyDB1dQg3uoQxMYIDzDK7w5j86L8+58LFoLTj5zDH/kfP4Akt+MxQ==</latexit><latexit sha1_base64="glsIYP92PSMcW8DkNNhZ8pvJRFs=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfRY9eKxBfsBbSib7aRdu9mE3Y1QQn+BFw+KePUnefPfuG1z0NYXFh7emWFn3iARXBvX/XYKa+sbm1vF7dLO7t7+QfnwqKXjVDFssljEqhNQjYJLbBpuBHYShTQKBLaD8d2s3n5CpXksH8wkQT+iQ8lDzqixVuOmX664VXcusgpeDhXIVe+Xv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtShphNrP5otOyZl1BiSMlX3SkLn7eyKjkdaTKLCdETUjvVybmf/VuqkJr/2MyyQ1KNniozAVxMRkdjUZcIXMiIkFyhS3uxI2oooyY7Mp2RC85ZNXoXVR9Sw3Liu12zyOIpzAKZyDB1dQg3uoQxMYIDzDK7w5j86L8+58LFoLTj5zDH/kfP4Akt+MxQ==</latexit><latexit sha1_base64="glsIYP92PSMcW8DkNNhZ8pvJRFs=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfRY9eKxBfsBbSib7aRdu9mE3Y1QQn+BFw+KePUnefPfuG1z0NYXFh7emWFn3iARXBvX/XYKa+sbm1vF7dLO7t7+QfnwqKXjVDFssljEqhNQjYJLbBpuBHYShTQKBLaD8d2s3n5CpXksH8wkQT+iQ8lDzqixVuOmX664VXcusgpeDhXIVe+Xv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtShphNrP5otOyZl1BiSMlX3SkLn7eyKjkdaTKLCdETUjvVybmf/VuqkJr/2MyyQ1KNniozAVxMRkdjUZcIXMiIkFyhS3uxI2oooyY7Mp2RC85ZNXoXVR9Sw3Liu12zyOIpzAKZyDB1dQg3uoQxMYIDzDK7w5j86L8+58LFoLTj5zDH/kfP4Akt+MxQ==</latexit><latexit sha1_base64="glsIYP92PSMcW8DkNNhZ8pvJRFs=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfRY9eKxBfsBbSib7aRdu9mE3Y1QQn+BFw+KePUnefPfuG1z0NYXFh7emWFn3iARXBvX/XYKa+sbm1vF7dLO7t7+QfnwqKXjVDFssljEqhNQjYJLbBpuBHYShTQKBLaD8d2s3n5CpXksH8wkQT+iQ8lDzqixVuOmX664VXcusgpeDhXIVe+Xv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtShphNrP5otOyZl1BiSMlX3SkLn7eyKjkdaTKLCdETUjvVybmf/VuqkJr/2MyyQ1KNniozAVxMRkdjUZcIXMiIkFyhS3uxI2oooyY7Mp2RC85ZNXoXVR9Sw3Liu12zyOIpzAKZyDB1dQg3uoQxMYIDzDK7w5j86L8+58LFoLTj5zDH/kfP4Akt+MxQ==</latexit>

Matrices- Matrix inverses are defined on square matrices

- Matrix inverse definition as a collection of properties:

- is full rank A<latexit sha1_base64="glsIYP92PSMcW8DkNNhZ8pvJRFs=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfRY9eKxBfsBbSib7aRdu9mE3Y1QQn+BFw+KePUnefPfuG1z0NYXFh7emWFn3iARXBvX/XYKa+sbm1vF7dLO7t7+QfnwqKXjVDFssljEqhNQjYJLbBpuBHYShTQKBLaD8d2s3n5CpXksH8wkQT+iQ8lDzqixVuOmX664VXcusgpeDhXIVe+Xv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtShphNrP5otOyZl1BiSMlX3SkLn7eyKjkdaTKLCdETUjvVybmf/VuqkJr/2MyyQ1KNniozAVxMRkdjUZcIXMiIkFyhS3uxI2oooyY7Mp2RC85ZNXoXVR9Sw3Liu12zyOIpzAKZyDB1dQg3uoQxMYIDzDK7w5j86L8+58LFoLTj5zDH/kfP4Akt+MxQ==</latexit><latexit sha1_base64="glsIYP92PSMcW8DkNNhZ8pvJRFs=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfRY9eKxBfsBbSib7aRdu9mE3Y1QQn+BFw+KePUnefPfuG1z0NYXFh7emWFn3iARXBvX/XYKa+sbm1vF7dLO7t7+QfnwqKXjVDFssljEqhNQjYJLbBpuBHYShTQKBLaD8d2s3n5CpXksH8wkQT+iQ8lDzqixVuOmX664VXcusgpeDhXIVe+Xv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtShphNrP5otOyZl1BiSMlX3SkLn7eyKjkdaTKLCdETUjvVybmf/VuqkJr/2MyyQ1KNniozAVxMRkdjUZcIXMiIkFyhS3uxI2oooyY7Mp2RC85ZNXoXVR9Sw3Liu12zyOIpzAKZyDB1dQg3uoQxMYIDzDK7w5j86L8+58LFoLTj5zDH/kfP4Akt+MxQ==</latexit><latexit sha1_base64="glsIYP92PSMcW8DkNNhZ8pvJRFs=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfRY9eKxBfsBbSib7aRdu9mE3Y1QQn+BFw+KePUnefPfuG1z0NYXFh7emWFn3iARXBvX/XYKa+sbm1vF7dLO7t7+QfnwqKXjVDFssljEqhNQjYJLbBpuBHYShTQKBLaD8d2s3n5CpXksH8wkQT+iQ8lDzqixVuOmX664VXcusgpeDhXIVe+Xv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtShphNrP5otOyZl1BiSMlX3SkLn7eyKjkdaTKLCdETUjvVybmf/VuqkJr/2MyyQ1KNniozAVxMRkdjUZcIXMiIkFyhS3uxI2oooyY7Mp2RC85ZNXoXVR9Sw3Liu12zyOIpzAKZyDB1dQg3uoQxMYIDzDK7w5j86L8+58LFoLTj5zDH/kfP4Akt+MxQ==</latexit><latexit sha1_base64="glsIYP92PSMcW8DkNNhZ8pvJRFs=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfRY9eKxBfsBbSib7aRdu9mE3Y1QQn+BFw+KePUnefPfuG1z0NYXFh7emWFn3iARXBvX/XYKa+sbm1vF7dLO7t7+QfnwqKXjVDFssljEqhNQjYJLbBpuBHYShTQKBLaD8d2s3n5CpXksH8wkQT+iQ8lDzqixVuOmX664VXcusgpeDhXIVe+Xv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtShphNrP5otOyZl1BiSMlX3SkLn7eyKjkdaTKLCdETUjvVybmf/VuqkJr/2MyyQ1KNniozAVxMRkdjUZcIXMiIkFyhS3uxI2oooyY7Mp2RC85ZNXoXVR9Sw3Liu12zyOIpzAKZyDB1dQg3uoQxMYIDzDK7w5j86L8+58LFoLTj5zDH/kfP4Akt+MxQ==</latexit>

- has empty nullspaceA<latexit sha1_base64="glsIYP92PSMcW8DkNNhZ8pvJRFs=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfRY9eKxBfsBbSib7aRdu9mE3Y1QQn+BFw+KePUnefPfuG1z0NYXFh7emWFn3iARXBvX/XYKa+sbm1vF7dLO7t7+QfnwqKXjVDFssljEqhNQjYJLbBpuBHYShTQKBLaD8d2s3n5CpXksH8wkQT+iQ8lDzqixVuOmX664VXcusgpeDhXIVe+Xv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtShphNrP5otOyZl1BiSMlX3SkLn7eyKjkdaTKLCdETUjvVybmf/VuqkJr/2MyyQ1KNniozAVxMRkdjUZcIXMiIkFyhS3uxI2oooyY7Mp2RC85ZNXoXVR9Sw3Liu12zyOIpzAKZyDB1dQg3uoQxMYIDzDK7w5j86L8+58LFoLTj5zDH/kfP4Akt+MxQ==</latexit><latexit sha1_base64="glsIYP92PSMcW8DkNNhZ8pvJRFs=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfRY9eKxBfsBbSib7aRdu9mE3Y1QQn+BFw+KePUnefPfuG1z0NYXFh7emWFn3iARXBvX/XYKa+sbm1vF7dLO7t7+QfnwqKXjVDFssljEqhNQjYJLbBpuBHYShTQKBLaD8d2s3n5CpXksH8wkQT+iQ8lDzqixVuOmX664VXcusgpeDhXIVe+Xv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtShphNrP5otOyZl1BiSMlX3SkLn7eyKjkdaTKLCdETUjvVybmf/VuqkJr/2MyyQ1KNniozAVxMRkdjUZcIXMiIkFyhS3uxI2oooyY7Mp2RC85ZNXoXVR9Sw3Liu12zyOIpzAKZyDB1dQg3uoQxMYIDzDK7w5j86L8+58LFoLTj5zDH/kfP4Akt+MxQ==</latexit><latexit sha1_base64="glsIYP92PSMcW8DkNNhZ8pvJRFs=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfRY9eKxBfsBbSib7aRdu9mE3Y1QQn+BFw+KePUnefPfuG1z0NYXFh7emWFn3iARXBvX/XYKa+sbm1vF7dLO7t7+QfnwqKXjVDFssljEqhNQjYJLbBpuBHYShTQKBLaD8d2s3n5CpXksH8wkQT+iQ8lDzqixVuOmX664VXcusgpeDhXIVe+Xv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtShphNrP5otOyZl1BiSMlX3SkLn7eyKjkdaTKLCdETUjvVybmf/VuqkJr/2MyyQ1KNniozAVxMRkdjUZcIXMiIkFyhS3uxI2oooyY7Mp2RC85ZNXoXVR9Sw3Liu12zyOIpzAKZyDB1dQg3uoQxMYIDzDK7w5j86L8+58LFoLTj5zDH/kfP4Akt+MxQ==</latexit><latexit sha1_base64="glsIYP92PSMcW8DkNNhZ8pvJRFs=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfRY9eKxBfsBbSib7aRdu9mE3Y1QQn+BFw+KePUnefPfuG1z0NYXFh7emWFn3iARXBvX/XYKa+sbm1vF7dLO7t7+QfnwqKXjVDFssljEqhNQjYJLbBpuBHYShTQKBLaD8d2s3n5CpXksH8wkQT+iQ8lDzqixVuOmX664VXcusgpeDhXIVe+Xv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtShphNrP5otOyZl1BiSMlX3SkLn7eyKjkdaTKLCdETUjvVybmf/VuqkJr/2MyyQ1KNniozAVxMRkdjUZcIXMiIkFyhS3uxI2oooyY7Mp2RC85ZNXoXVR9Sw3Liu12zyOIpzAKZyDB1dQg3uoQxMYIDzDK7w5j86L8+58LFoLTj5zDH/kfP4Akt+MxQ==</latexit>

- The equation has only the trivial solution Ax = 0<latexit sha1_base64="yY1xCWv4apZjphe88GtRt86JNMg=">AAAB7XicbZDLSgMxFIZPvNZ6q7p0EyyCqzIjgm6EqhuXFewF2qFk0kwbm0mGJCOWoe/gxoUibn0fd76NaTsLbf0h8PGfc8g5f5gIbqznfaOl5ZXVtfXCRnFza3tnt7S33zAq1ZTVqRJKt0JimOCS1S23grUSzUgcCtYMhzeTevORacOVvLejhAUx6UsecUqssxpXT/gSe91S2at4U+FF8HMoQ65at/TV6SmaxkxaKogxbd9LbJARbTkVbFzspIYlhA5Jn7UdShIzE2TTbcf42Dk9HCntnrR46v6eyEhszCgOXWdM7MDM1ybmf7V2aqOLIOMySS2TdPZRlApsFZ6cjntcM2rFyAGhmrtdMR0QTah1ARVdCP78yYvQOK34ju/OytXrPI4CHMIRnIAP51CFW6hBHSg8wDO8whtS6AW9o49Z6xLKZw7gj9DnDwvHjhw=</latexit><latexit sha1_base64="yY1xCWv4apZjphe88GtRt86JNMg=">AAAB7XicbZDLSgMxFIZPvNZ6q7p0EyyCqzIjgm6EqhuXFewF2qFk0kwbm0mGJCOWoe/gxoUibn0fd76NaTsLbf0h8PGfc8g5f5gIbqznfaOl5ZXVtfXCRnFza3tnt7S33zAq1ZTVqRJKt0JimOCS1S23grUSzUgcCtYMhzeTevORacOVvLejhAUx6UsecUqssxpXT/gSe91S2at4U+FF8HMoQ65at/TV6SmaxkxaKogxbd9LbJARbTkVbFzspIYlhA5Jn7UdShIzE2TTbcf42Dk9HCntnrR46v6eyEhszCgOXWdM7MDM1ybmf7V2aqOLIOMySS2TdPZRlApsFZ6cjntcM2rFyAGhmrtdMR0QTah1ARVdCP78yYvQOK34ju/OytXrPI4CHMIRnIAP51CFW6hBHSg8wDO8whtS6AW9o49Z6xLKZw7gj9DnDwvHjhw=</latexit><latexit sha1_base64="yY1xCWv4apZjphe88GtRt86JNMg=">AAAB7XicbZDLSgMxFIZPvNZ6q7p0EyyCqzIjgm6EqhuXFewF2qFk0kwbm0mGJCOWoe/gxoUibn0fd76NaTsLbf0h8PGfc8g5f5gIbqznfaOl5ZXVtfXCRnFza3tnt7S33zAq1ZTVqRJKt0JimOCS1S23grUSzUgcCtYMhzeTevORacOVvLejhAUx6UsecUqssxpXT/gSe91S2at4U+FF8HMoQ65at/TV6SmaxkxaKogxbd9LbJARbTkVbFzspIYlhA5Jn7UdShIzE2TTbcf42Dk9HCntnrR46v6eyEhszCgOXWdM7MDM1ybmf7V2aqOLIOMySS2TdPZRlApsFZ6cjntcM2rFyAGhmrtdMR0QTah1ARVdCP78yYvQOK34ju/OytXrPI4CHMIRnIAP51CFW6hBHSg8wDO8whtS6AW9o49Z6xLKZw7gj9DnDwvHjhw=</latexit><latexit sha1_base64="yY1xCWv4apZjphe88GtRt86JNMg=">AAAB7XicbZDLSgMxFIZPvNZ6q7p0EyyCqzIjgm6EqhuXFewF2qFk0kwbm0mGJCOWoe/gxoUibn0fd76NaTsLbf0h8PGfc8g5f5gIbqznfaOl5ZXVtfXCRnFza3tnt7S33zAq1ZTVqRJKt0JimOCS1S23grUSzUgcCtYMhzeTevORacOVvLejhAUx6UsecUqssxpXT/gSe91S2at4U+FF8HMoQ65at/TV6SmaxkxaKogxbd9LbJARbTkVbFzspIYlhA5Jn7UdShIzE2TTbcf42Dk9HCntnrR46v6eyEhszCgOXWdM7MDM1ybmf7V2aqOLIOMySS2TdPZRlApsFZ6cjntcM2rFyAGhmrtdMR0QTah1ARVdCP78yYvQOK34ju/OytXrPI4CHMIRnIAP51CFW6hBHSg8wDO8whtS6AW9o49Z6xLKZw7gj9DnDwvHjhw=</latexit>

x = 0<latexit sha1_base64="TMIHVM7cjPNIpSJEQSU+/pH7FkY=">AAAB7HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfQiFL14rGBqoQ1ls520SzebsLsRS+hv8OJBEa/+IG/+G7dtDtr6wsLDOzPszBumgmvjut9OaWV1bX2jvFnZ2t7Z3avuH7R0kimGPktEotoh1Si4RN9wI7CdKqRxKPAhHN1M6w+PqDRP5L0ZpxjEdCB5xBk11vKfyBVxe9WaW3dnIsvgFVCDQs1e9avbT1gWozRMUK07npuaIKfKcCZwUulmGlPKRnSAHYuSxqiDfLbshJxYp0+iRNknDZm5vydyGms9jkPbGVMz1Iu1qflfrZOZ6DLIuUwzg5LNP4oyQUxCppeTPlfIjBhboExxuythQ6ooMzafig3BWzx5GVpndc/y3XmtcV3EUYYjOIZT8OACGnALTfCBAYdneIU3RzovzrvzMW8tOcXMIfyR8/kDgwON0Q==</latexit><latexit sha1_base64="TMIHVM7cjPNIpSJEQSU+/pH7FkY=">AAAB7HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfQiFL14rGBqoQ1ls520SzebsLsRS+hv8OJBEa/+IG/+G7dtDtr6wsLDOzPszBumgmvjut9OaWV1bX2jvFnZ2t7Z3avuH7R0kimGPktEotoh1Si4RN9wI7CdKqRxKPAhHN1M6w+PqDRP5L0ZpxjEdCB5xBk11vKfyBVxe9WaW3dnIsvgFVCDQs1e9avbT1gWozRMUK07npuaIKfKcCZwUulmGlPKRnSAHYuSxqiDfLbshJxYp0+iRNknDZm5vydyGms9jkPbGVMz1Iu1qflfrZOZ6DLIuUwzg5LNP4oyQUxCppeTPlfIjBhboExxuythQ6ooMzafig3BWzx5GVpndc/y3XmtcV3EUYYjOIZT8OACGnALTfCBAYdneIU3RzovzrvzMW8tOcXMIfyR8/kDgwON0Q==</latexit><latexit sha1_base64="TMIHVM7cjPNIpSJEQSU+/pH7FkY=">AAAB7HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfQiFL14rGBqoQ1ls520SzebsLsRS+hv8OJBEa/+IG/+G7dtDtr6wsLDOzPszBumgmvjut9OaWV1bX2jvFnZ2t7Z3avuH7R0kimGPktEotoh1Si4RN9wI7CdKqRxKPAhHN1M6w+PqDRP5L0ZpxjEdCB5xBk11vKfyBVxe9WaW3dnIsvgFVCDQs1e9avbT1gWozRMUK07npuaIKfKcCZwUulmGlPKRnSAHYuSxqiDfLbshJxYp0+iRNknDZm5vydyGms9jkPbGVMz1Iu1qflfrZOZ6DLIuUwzg5LNP4oyQUxCppeTPlfIjBhboExxuythQ6ooMzafig3BWzx5GVpndc/y3XmtcV3EUYYjOIZT8OACGnALTfCBAYdneIU3RzovzrvzMW8tOcXMIfyR8/kDgwON0Q==</latexit><latexit sha1_base64="TMIHVM7cjPNIpSJEQSU+/pH7FkY=">AAAB7HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfQiFL14rGBqoQ1ls520SzebsLsRS+hv8OJBEa/+IG/+G7dtDtr6wsLDOzPszBumgmvjut9OaWV1bX2jvFnZ2t7Z3avuH7R0kimGPktEotoh1Si4RN9wI7CdKqRxKPAhHN1M6w+PqDRP5L0ZpxjEdCB5xBk11vKfyBVxe9WaW3dnIsvgFVCDQs1e9avbT1gWozRMUK07npuaIKfKcCZwUulmGlPKRnSAHYuSxqiDfLbshJxYp0+iRNknDZm5vydyGms9jkPbGVMz1Iu1qflfrZOZ6DLIuUwzg5LNP4oyQUxCppeTPlfIjBhboExxuythQ6ooMzafig3BWzx5GVpndc/y3XmtcV3EUYYjOIZT8OACGnALTfCBAYdneIU3RzovzrvzMW8tOcXMIfyR8/kDgwON0Q==</latexit>

- The linear system has a unique solutionAx = b<latexit sha1_base64="8Viwh2D00yJ5mcgJngUqqpm71Gs=">AAAB7XicbZDLSgMxFIZPvNZ6q7p0EyyCqzIjgm6EqhuXFewF2qFk0kwbm0mGJCOWoe/gxoUibn0fd76NaTsLbf0h8PGfc8g5f5gIbqznfaOl5ZXVtfXCRnFza3tnt7S33zAq1ZTVqRJKt0JimOCS1S23grUSzUgcCtYMhzeTevORacOVvLejhAUx6UsecUqssxpXT/gSh91S2at4U+FF8HMoQ65at/TV6SmaxkxaKogxbd9LbJARbTkVbFzspIYlhA5Jn7UdShIzE2TTbcf42Dk9HCntnrR46v6eyEhszCgOXWdM7MDM1ybmf7V2aqOLIOMySS2TdPZRlApsFZ6cjntcM2rFyAGhmrtdMR0QTah1ARVdCP78yYvQOK34ju/OytXrPI4CHMIRnIAP51CFW6hBHSg8wDO8whtS6AW9o49Z6xLKZw7gj9DnD1ePjk4=</latexit><latexit sha1_base64="8Viwh2D00yJ5mcgJngUqqpm71Gs=">AAAB7XicbZDLSgMxFIZPvNZ6q7p0EyyCqzIjgm6EqhuXFewF2qFk0kwbm0mGJCOWoe/gxoUibn0fd76NaTsLbf0h8PGfc8g5f5gIbqznfaOl5ZXVtfXCRnFza3tnt7S33zAq1ZTVqRJKt0JimOCS1S23grUSzUgcCtYMhzeTevORacOVvLejhAUx6UsecUqssxpXT/gSh91S2at4U+FF8HMoQ65at/TV6SmaxkxaKogxbd9LbJARbTkVbFzspIYlhA5Jn7UdShIzE2TTbcf42Dk9HCntnrR46v6eyEhszCgOXWdM7MDM1ybmf7V2aqOLIOMySS2TdPZRlApsFZ6cjntcM2rFyAGhmrtdMR0QTah1ARVdCP78yYvQOK34ju/OytXrPI4CHMIRnIAP51CFW6hBHSg8wDO8whtS6AW9o49Z6xLKZw7gj9DnD1ePjk4=</latexit><latexit sha1_base64="8Viwh2D00yJ5mcgJngUqqpm71Gs=">AAAB7XicbZDLSgMxFIZPvNZ6q7p0EyyCqzIjgm6EqhuXFewF2qFk0kwbm0mGJCOWoe/gxoUibn0fd76NaTsLbf0h8PGfc8g5f5gIbqznfaOl5ZXVtfXCRnFza3tnt7S33zAq1ZTVqRJKt0JimOCS1S23grUSzUgcCtYMhzeTevORacOVvLejhAUx6UsecUqssxpXT/gSh91S2at4U+FF8HMoQ65at/TV6SmaxkxaKogxbd9LbJARbTkVbFzspIYlhA5Jn7UdShIzE2TTbcf42Dk9HCntnrR46v6eyEhszCgOXWdM7MDM1ybmf7V2aqOLIOMySS2TdPZRlApsFZ6cjntcM2rFyAGhmrtdMR0QTah1ARVdCP78yYvQOK34ju/OytXrPI4CHMIRnIAP51CFW6hBHSg8wDO8whtS6AW9o49Z6xLKZw7gj9DnD1ePjk4=</latexit><latexit sha1_base64="8Viwh2D00yJ5mcgJngUqqpm71Gs=">AAAB7XicbZDLSgMxFIZPvNZ6q7p0EyyCqzIjgm6EqhuXFewF2qFk0kwbm0mGJCOWoe/gxoUibn0fd76NaTsLbf0h8PGfc8g5f5gIbqznfaOl5ZXVtfXCRnFza3tnt7S33zAq1ZTVqRJKt0JimOCS1S23grUSzUgcCtYMhzeTevORacOVvLejhAUx6UsecUqssxpXT/gSh91S2at4U+FF8HMoQ65at/TV6SmaxkxaKogxbd9LbJARbTkVbFzspIYlhA5Jn7UdShIzE2TTbcf42Dk9HCntnrR46v6eyEhszCgOXWdM7MDM1ybmf7V2aqOLIOMySS2TdPZRlApsFZ6cjntcM2rFyAGhmrtdMR0QTah1ARVdCP78yYvQOK34ju/OytXrPI4CHMIRnIAP51CFW6hBHSg8wDO8whtS6AW9o49Z6xLKZw7gj9DnD1ePjk4=</latexit>

Page 145: COMP 414/514: Optimization - Algorithms, Complexity and

- The columns and rows of are linearly independentA<latexit sha1_base64="glsIYP92PSMcW8DkNNhZ8pvJRFs=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfRY9eKxBfsBbSib7aRdu9mE3Y1QQn+BFw+KePUnefPfuG1z0NYXFh7emWFn3iARXBvX/XYKa+sbm1vF7dLO7t7+QfnwqKXjVDFssljEqhNQjYJLbBpuBHYShTQKBLaD8d2s3n5CpXksH8wkQT+iQ8lDzqixVuOmX664VXcusgpeDhXIVe+Xv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtShphNrP5otOyZl1BiSMlX3SkLn7eyKjkdaTKLCdETUjvVybmf/VuqkJr/2MyyQ1KNniozAVxMRkdjUZcIXMiIkFyhS3uxI2oooyY7Mp2RC85ZNXoXVR9Sw3Liu12zyOIpzAKZyDB1dQg3uoQxMYIDzDK7w5j86L8+58LFoLTj5zDH/kfP4Akt+MxQ==</latexit><latexit sha1_base64="glsIYP92PSMcW8DkNNhZ8pvJRFs=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfRY9eKxBfsBbSib7aRdu9mE3Y1QQn+BFw+KePUnefPfuG1z0NYXFh7emWFn3iARXBvX/XYKa+sbm1vF7dLO7t7+QfnwqKXjVDFssljEqhNQjYJLbBpuBHYShTQKBLaD8d2s3n5CpXksH8wkQT+iQ8lDzqixVuOmX664VXcusgpeDhXIVe+Xv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtShphNrP5otOyZl1BiSMlX3SkLn7eyKjkdaTKLCdETUjvVybmf/VuqkJr/2MyyQ1KNniozAVxMRkdjUZcIXMiIkFyhS3uxI2oooyY7Mp2RC85ZNXoXVR9Sw3Liu12zyOIpzAKZyDB1dQg3uoQxMYIDzDK7w5j86L8+58LFoLTj5zDH/kfP4Akt+MxQ==</latexit><latexit sha1_base64="glsIYP92PSMcW8DkNNhZ8pvJRFs=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfRY9eKxBfsBbSib7aRdu9mE3Y1QQn+BFw+KePUnefPfuG1z0NYXFh7emWFn3iARXBvX/XYKa+sbm1vF7dLO7t7+QfnwqKXjVDFssljEqhNQjYJLbBpuBHYShTQKBLaD8d2s3n5CpXksH8wkQT+iQ8lDzqixVuOmX664VXcusgpeDhXIVe+Xv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtShphNrP5otOyZl1BiSMlX3SkLn7eyKjkdaTKLCdETUjvVybmf/VuqkJr/2MyyQ1KNniozAVxMRkdjUZcIXMiIkFyhS3uxI2oooyY7Mp2RC85ZNXoXVR9Sw3Liu12zyOIpzAKZyDB1dQg3uoQxMYIDzDK7w5j86L8+58LFoLTj5zDH/kfP4Akt+MxQ==</latexit><latexit sha1_base64="glsIYP92PSMcW8DkNNhZ8pvJRFs=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfRY9eKxBfsBbSib7aRdu9mE3Y1QQn+BFw+KePUnefPfuG1z0NYXFh7emWFn3iARXBvX/XYKa+sbm1vF7dLO7t7+QfnwqKXjVDFssljEqhNQjYJLbBpuBHYShTQKBLaD8d2s3n5CpXksH8wkQT+iQ8lDzqixVuOmX664VXcusgpeDhXIVe+Xv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtShphNrP5otOyZl1BiSMlX3SkLn7eyKjkdaTKLCdETUjvVybmf/VuqkJr/2MyyQ1KNniozAVxMRkdjUZcIXMiIkFyhS3uxI2oooyY7Mp2RC85ZNXoXVR9Sw3Liu12zyOIpzAKZyDB1dQg3uoQxMYIDzDK7w5j86L8+58LFoLTj5zDH/kfP4Akt+MxQ==</latexit>

Matrices- Matrix inverses are defined on square matrices

- Matrix inverse definition as a collection of properties:

- is full rank A<latexit sha1_base64="glsIYP92PSMcW8DkNNhZ8pvJRFs=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfRY9eKxBfsBbSib7aRdu9mE3Y1QQn+BFw+KePUnefPfuG1z0NYXFh7emWFn3iARXBvX/XYKa+sbm1vF7dLO7t7+QfnwqKXjVDFssljEqhNQjYJLbBpuBHYShTQKBLaD8d2s3n5CpXksH8wkQT+iQ8lDzqixVuOmX664VXcusgpeDhXIVe+Xv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtShphNrP5otOyZl1BiSMlX3SkLn7eyKjkdaTKLCdETUjvVybmf/VuqkJr/2MyyQ1KNniozAVxMRkdjUZcIXMiIkFyhS3uxI2oooyY7Mp2RC85ZNXoXVR9Sw3Liu12zyOIpzAKZyDB1dQg3uoQxMYIDzDK7w5j86L8+58LFoLTj5zDH/kfP4Akt+MxQ==</latexit><latexit sha1_base64="glsIYP92PSMcW8DkNNhZ8pvJRFs=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfRY9eKxBfsBbSib7aRdu9mE3Y1QQn+BFw+KePUnefPfuG1z0NYXFh7emWFn3iARXBvX/XYKa+sbm1vF7dLO7t7+QfnwqKXjVDFssljEqhNQjYJLbBpuBHYShTQKBLaD8d2s3n5CpXksH8wkQT+iQ8lDzqixVuOmX664VXcusgpeDhXIVe+Xv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtShphNrP5otOyZl1BiSMlX3SkLn7eyKjkdaTKLCdETUjvVybmf/VuqkJr/2MyyQ1KNniozAVxMRkdjUZcIXMiIkFyhS3uxI2oooyY7Mp2RC85ZNXoXVR9Sw3Liu12zyOIpzAKZyDB1dQg3uoQxMYIDzDK7w5j86L8+58LFoLTj5zDH/kfP4Akt+MxQ==</latexit><latexit sha1_base64="glsIYP92PSMcW8DkNNhZ8pvJRFs=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfRY9eKxBfsBbSib7aRdu9mE3Y1QQn+BFw+KePUnefPfuG1z0NYXFh7emWFn3iARXBvX/XYKa+sbm1vF7dLO7t7+QfnwqKXjVDFssljEqhNQjYJLbBpuBHYShTQKBLaD8d2s3n5CpXksH8wkQT+iQ8lDzqixVuOmX664VXcusgpeDhXIVe+Xv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtShphNrP5otOyZl1BiSMlX3SkLn7eyKjkdaTKLCdETUjvVybmf/VuqkJr/2MyyQ1KNniozAVxMRkdjUZcIXMiIkFyhS3uxI2oooyY7Mp2RC85ZNXoXVR9Sw3Liu12zyOIpzAKZyDB1dQg3uoQxMYIDzDK7w5j86L8+58LFoLTj5zDH/kfP4Akt+MxQ==</latexit><latexit sha1_base64="glsIYP92PSMcW8DkNNhZ8pvJRFs=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfRY9eKxBfsBbSib7aRdu9mE3Y1QQn+BFw+KePUnefPfuG1z0NYXFh7emWFn3iARXBvX/XYKa+sbm1vF7dLO7t7+QfnwqKXjVDFssljEqhNQjYJLbBpuBHYShTQKBLaD8d2s3n5CpXksH8wkQT+iQ8lDzqixVuOmX664VXcusgpeDhXIVe+Xv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtShphNrP5otOyZl1BiSMlX3SkLn7eyKjkdaTKLCdETUjvVybmf/VuqkJr/2MyyQ1KNniozAVxMRkdjUZcIXMiIkFyhS3uxI2oooyY7Mp2RC85ZNXoXVR9Sw3Liu12zyOIpzAKZyDB1dQg3uoQxMYIDzDK7w5j86L8+58LFoLTj5zDH/kfP4Akt+MxQ==</latexit>

- has empty nullspaceA<latexit sha1_base64="glsIYP92PSMcW8DkNNhZ8pvJRFs=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfRY9eKxBfsBbSib7aRdu9mE3Y1QQn+BFw+KePUnefPfuG1z0NYXFh7emWFn3iARXBvX/XYKa+sbm1vF7dLO7t7+QfnwqKXjVDFssljEqhNQjYJLbBpuBHYShTQKBLaD8d2s3n5CpXksH8wkQT+iQ8lDzqixVuOmX664VXcusgpeDhXIVe+Xv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtShphNrP5otOyZl1BiSMlX3SkLn7eyKjkdaTKLCdETUjvVybmf/VuqkJr/2MyyQ1KNniozAVxMRkdjUZcIXMiIkFyhS3uxI2oooyY7Mp2RC85ZNXoXVR9Sw3Liu12zyOIpzAKZyDB1dQg3uoQxMYIDzDK7w5j86L8+58LFoLTj5zDH/kfP4Akt+MxQ==</latexit><latexit sha1_base64="glsIYP92PSMcW8DkNNhZ8pvJRFs=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfRY9eKxBfsBbSib7aRdu9mE3Y1QQn+BFw+KePUnefPfuG1z0NYXFh7emWFn3iARXBvX/XYKa+sbm1vF7dLO7t7+QfnwqKXjVDFssljEqhNQjYJLbBpuBHYShTQKBLaD8d2s3n5CpXksH8wkQT+iQ8lDzqixVuOmX664VXcusgpeDhXIVe+Xv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtShphNrP5otOyZl1BiSMlX3SkLn7eyKjkdaTKLCdETUjvVybmf/VuqkJr/2MyyQ1KNniozAVxMRkdjUZcIXMiIkFyhS3uxI2oooyY7Mp2RC85ZNXoXVR9Sw3Liu12zyOIpzAKZyDB1dQg3uoQxMYIDzDK7w5j86L8+58LFoLTj5zDH/kfP4Akt+MxQ==</latexit><latexit sha1_base64="glsIYP92PSMcW8DkNNhZ8pvJRFs=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfRY9eKxBfsBbSib7aRdu9mE3Y1QQn+BFw+KePUnefPfuG1z0NYXFh7emWFn3iARXBvX/XYKa+sbm1vF7dLO7t7+QfnwqKXjVDFssljEqhNQjYJLbBpuBHYShTQKBLaD8d2s3n5CpXksH8wkQT+iQ8lDzqixVuOmX664VXcusgpeDhXIVe+Xv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtShphNrP5otOyZl1BiSMlX3SkLn7eyKjkdaTKLCdETUjvVybmf/VuqkJr/2MyyQ1KNniozAVxMRkdjUZcIXMiIkFyhS3uxI2oooyY7Mp2RC85ZNXoXVR9Sw3Liu12zyOIpzAKZyDB1dQg3uoQxMYIDzDK7w5j86L8+58LFoLTj5zDH/kfP4Akt+MxQ==</latexit><latexit sha1_base64="glsIYP92PSMcW8DkNNhZ8pvJRFs=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfRY9eKxBfsBbSib7aRdu9mE3Y1QQn+BFw+KePUnefPfuG1z0NYXFh7emWFn3iARXBvX/XYKa+sbm1vF7dLO7t7+QfnwqKXjVDFssljEqhNQjYJLbBpuBHYShTQKBLaD8d2s3n5CpXksH8wkQT+iQ8lDzqixVuOmX664VXcusgpeDhXIVe+Xv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtShphNrP5otOyZl1BiSMlX3SkLn7eyKjkdaTKLCdETUjvVybmf/VuqkJr/2MyyQ1KNniozAVxMRkdjUZcIXMiIkFyhS3uxI2oooyY7Mp2RC85ZNXoXVR9Sw3Liu12zyOIpzAKZyDB1dQg3uoQxMYIDzDK7w5j86L8+58LFoLTj5zDH/kfP4Akt+MxQ==</latexit>

- The equation has only the trivial solution Ax = 0<latexit sha1_base64="yY1xCWv4apZjphe88GtRt86JNMg=">AAAB7XicbZDLSgMxFIZPvNZ6q7p0EyyCqzIjgm6EqhuXFewF2qFk0kwbm0mGJCOWoe/gxoUibn0fd76NaTsLbf0h8PGfc8g5f5gIbqznfaOl5ZXVtfXCRnFza3tnt7S33zAq1ZTVqRJKt0JimOCS1S23grUSzUgcCtYMhzeTevORacOVvLejhAUx6UsecUqssxpXT/gSe91S2at4U+FF8HMoQ65at/TV6SmaxkxaKogxbd9LbJARbTkVbFzspIYlhA5Jn7UdShIzE2TTbcf42Dk9HCntnrR46v6eyEhszCgOXWdM7MDM1ybmf7V2aqOLIOMySS2TdPZRlApsFZ6cjntcM2rFyAGhmrtdMR0QTah1ARVdCP78yYvQOK34ju/OytXrPI4CHMIRnIAP51CFW6hBHSg8wDO8whtS6AW9o49Z6xLKZw7gj9DnDwvHjhw=</latexit><latexit sha1_base64="yY1xCWv4apZjphe88GtRt86JNMg=">AAAB7XicbZDLSgMxFIZPvNZ6q7p0EyyCqzIjgm6EqhuXFewF2qFk0kwbm0mGJCOWoe/gxoUibn0fd76NaTsLbf0h8PGfc8g5f5gIbqznfaOl5ZXVtfXCRnFza3tnt7S33zAq1ZTVqRJKt0JimOCS1S23grUSzUgcCtYMhzeTevORacOVvLejhAUx6UsecUqssxpXT/gSe91S2at4U+FF8HMoQ65at/TV6SmaxkxaKogxbd9LbJARbTkVbFzspIYlhA5Jn7UdShIzE2TTbcf42Dk9HCntnrR46v6eyEhszCgOXWdM7MDM1ybmf7V2aqOLIOMySS2TdPZRlApsFZ6cjntcM2rFyAGhmrtdMR0QTah1ARVdCP78yYvQOK34ju/OytXrPI4CHMIRnIAP51CFW6hBHSg8wDO8whtS6AW9o49Z6xLKZw7gj9DnDwvHjhw=</latexit><latexit sha1_base64="yY1xCWv4apZjphe88GtRt86JNMg=">AAAB7XicbZDLSgMxFIZPvNZ6q7p0EyyCqzIjgm6EqhuXFewF2qFk0kwbm0mGJCOWoe/gxoUibn0fd76NaTsLbf0h8PGfc8g5f5gIbqznfaOl5ZXVtfXCRnFza3tnt7S33zAq1ZTVqRJKt0JimOCS1S23grUSzUgcCtYMhzeTevORacOVvLejhAUx6UsecUqssxpXT/gSe91S2at4U+FF8HMoQ65at/TV6SmaxkxaKogxbd9LbJARbTkVbFzspIYlhA5Jn7UdShIzE2TTbcf42Dk9HCntnrR46v6eyEhszCgOXWdM7MDM1ybmf7V2aqOLIOMySS2TdPZRlApsFZ6cjntcM2rFyAGhmrtdMR0QTah1ARVdCP78yYvQOK34ju/OytXrPI4CHMIRnIAP51CFW6hBHSg8wDO8whtS6AW9o49Z6xLKZw7gj9DnDwvHjhw=</latexit><latexit sha1_base64="yY1xCWv4apZjphe88GtRt86JNMg=">AAAB7XicbZDLSgMxFIZPvNZ6q7p0EyyCqzIjgm6EqhuXFewF2qFk0kwbm0mGJCOWoe/gxoUibn0fd76NaTsLbf0h8PGfc8g5f5gIbqznfaOl5ZXVtfXCRnFza3tnt7S33zAq1ZTVqRJKt0JimOCS1S23grUSzUgcCtYMhzeTevORacOVvLejhAUx6UsecUqssxpXT/gSe91S2at4U+FF8HMoQ65at/TV6SmaxkxaKogxbd9LbJARbTkVbFzspIYlhA5Jn7UdShIzE2TTbcf42Dk9HCntnrR46v6eyEhszCgOXWdM7MDM1ybmf7V2aqOLIOMySS2TdPZRlApsFZ6cjntcM2rFyAGhmrtdMR0QTah1ARVdCP78yYvQOK34ju/OytXrPI4CHMIRnIAP51CFW6hBHSg8wDO8whtS6AW9o49Z6xLKZw7gj9DnDwvHjhw=</latexit>

x = 0<latexit sha1_base64="TMIHVM7cjPNIpSJEQSU+/pH7FkY=">AAAB7HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfQiFL14rGBqoQ1ls520SzebsLsRS+hv8OJBEa/+IG/+G7dtDtr6wsLDOzPszBumgmvjut9OaWV1bX2jvFnZ2t7Z3avuH7R0kimGPktEotoh1Si4RN9wI7CdKqRxKPAhHN1M6w+PqDRP5L0ZpxjEdCB5xBk11vKfyBVxe9WaW3dnIsvgFVCDQs1e9avbT1gWozRMUK07npuaIKfKcCZwUulmGlPKRnSAHYuSxqiDfLbshJxYp0+iRNknDZm5vydyGms9jkPbGVMz1Iu1qflfrZOZ6DLIuUwzg5LNP4oyQUxCppeTPlfIjBhboExxuythQ6ooMzafig3BWzx5GVpndc/y3XmtcV3EUYYjOIZT8OACGnALTfCBAYdneIU3RzovzrvzMW8tOcXMIfyR8/kDgwON0Q==</latexit><latexit sha1_base64="TMIHVM7cjPNIpSJEQSU+/pH7FkY=">AAAB7HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfQiFL14rGBqoQ1ls520SzebsLsRS+hv8OJBEa/+IG/+G7dtDtr6wsLDOzPszBumgmvjut9OaWV1bX2jvFnZ2t7Z3avuH7R0kimGPktEotoh1Si4RN9wI7CdKqRxKPAhHN1M6w+PqDRP5L0ZpxjEdCB5xBk11vKfyBVxe9WaW3dnIsvgFVCDQs1e9avbT1gWozRMUK07npuaIKfKcCZwUulmGlPKRnSAHYuSxqiDfLbshJxYp0+iRNknDZm5vydyGms9jkPbGVMz1Iu1qflfrZOZ6DLIuUwzg5LNP4oyQUxCppeTPlfIjBhboExxuythQ6ooMzafig3BWzx5GVpndc/y3XmtcV3EUYYjOIZT8OACGnALTfCBAYdneIU3RzovzrvzMW8tOcXMIfyR8/kDgwON0Q==</latexit><latexit sha1_base64="TMIHVM7cjPNIpSJEQSU+/pH7FkY=">AAAB7HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfQiFL14rGBqoQ1ls520SzebsLsRS+hv8OJBEa/+IG/+G7dtDtr6wsLDOzPszBumgmvjut9OaWV1bX2jvFnZ2t7Z3avuH7R0kimGPktEotoh1Si4RN9wI7CdKqRxKPAhHN1M6w+PqDRP5L0ZpxjEdCB5xBk11vKfyBVxe9WaW3dnIsvgFVCDQs1e9avbT1gWozRMUK07npuaIKfKcCZwUulmGlPKRnSAHYuSxqiDfLbshJxYp0+iRNknDZm5vydyGms9jkPbGVMz1Iu1qflfrZOZ6DLIuUwzg5LNP4oyQUxCppeTPlfIjBhboExxuythQ6ooMzafig3BWzx5GVpndc/y3XmtcV3EUYYjOIZT8OACGnALTfCBAYdneIU3RzovzrvzMW8tOcXMIfyR8/kDgwON0Q==</latexit><latexit sha1_base64="TMIHVM7cjPNIpSJEQSU+/pH7FkY=">AAAB7HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfQiFL14rGBqoQ1ls520SzebsLsRS+hv8OJBEa/+IG/+G7dtDtr6wsLDOzPszBumgmvjut9OaWV1bX2jvFnZ2t7Z3avuH7R0kimGPktEotoh1Si4RN9wI7CdKqRxKPAhHN1M6w+PqDRP5L0ZpxjEdCB5xBk11vKfyBVxe9WaW3dnIsvgFVCDQs1e9avbT1gWozRMUK07npuaIKfKcCZwUulmGlPKRnSAHYuSxqiDfLbshJxYp0+iRNknDZm5vydyGms9jkPbGVMz1Iu1qflfrZOZ6DLIuUwzg5LNP4oyQUxCppeTPlfIjBhboExxuythQ6ooMzafig3BWzx5GVpndc/y3XmtcV3EUYYjOIZT8OACGnALTfCBAYdneIU3RzovzrvzMW8tOcXMIfyR8/kDgwON0Q==</latexit>

- The linear system has a unique solutionAx = b<latexit sha1_base64="8Viwh2D00yJ5mcgJngUqqpm71Gs=">AAAB7XicbZDLSgMxFIZPvNZ6q7p0EyyCqzIjgm6EqhuXFewF2qFk0kwbm0mGJCOWoe/gxoUibn0fd76NaTsLbf0h8PGfc8g5f5gIbqznfaOl5ZXVtfXCRnFza3tnt7S33zAq1ZTVqRJKt0JimOCS1S23grUSzUgcCtYMhzeTevORacOVvLejhAUx6UsecUqssxpXT/gSh91S2at4U+FF8HMoQ65at/TV6SmaxkxaKogxbd9LbJARbTkVbFzspIYlhA5Jn7UdShIzE2TTbcf42Dk9HCntnrR46v6eyEhszCgOXWdM7MDM1ybmf7V2aqOLIOMySS2TdPZRlApsFZ6cjntcM2rFyAGhmrtdMR0QTah1ARVdCP78yYvQOK34ju/OytXrPI4CHMIRnIAP51CFW6hBHSg8wDO8whtS6AW9o49Z6xLKZw7gj9DnD1ePjk4=</latexit><latexit sha1_base64="8Viwh2D00yJ5mcgJngUqqpm71Gs=">AAAB7XicbZDLSgMxFIZPvNZ6q7p0EyyCqzIjgm6EqhuXFewF2qFk0kwbm0mGJCOWoe/gxoUibn0fd76NaTsLbf0h8PGfc8g5f5gIbqznfaOl5ZXVtfXCRnFza3tnt7S33zAq1ZTVqRJKt0JimOCS1S23grUSzUgcCtYMhzeTevORacOVvLejhAUx6UsecUqssxpXT/gSh91S2at4U+FF8HMoQ65at/TV6SmaxkxaKogxbd9LbJARbTkVbFzspIYlhA5Jn7UdShIzE2TTbcf42Dk9HCntnrR46v6eyEhszCgOXWdM7MDM1ybmf7V2aqOLIOMySS2TdPZRlApsFZ6cjntcM2rFyAGhmrtdMR0QTah1ARVdCP78yYvQOK34ju/OytXrPI4CHMIRnIAP51CFW6hBHSg8wDO8whtS6AW9o49Z6xLKZw7gj9DnD1ePjk4=</latexit><latexit sha1_base64="8Viwh2D00yJ5mcgJngUqqpm71Gs=">AAAB7XicbZDLSgMxFIZPvNZ6q7p0EyyCqzIjgm6EqhuXFewF2qFk0kwbm0mGJCOWoe/gxoUibn0fd76NaTsLbf0h8PGfc8g5f5gIbqznfaOl5ZXVtfXCRnFza3tnt7S33zAq1ZTVqRJKt0JimOCS1S23grUSzUgcCtYMhzeTevORacOVvLejhAUx6UsecUqssxpXT/gSh91S2at4U+FF8HMoQ65at/TV6SmaxkxaKogxbd9LbJARbTkVbFzspIYlhA5Jn7UdShIzE2TTbcf42Dk9HCntnrR46v6eyEhszCgOXWdM7MDM1ybmf7V2aqOLIOMySS2TdPZRlApsFZ6cjntcM2rFyAGhmrtdMR0QTah1ARVdCP78yYvQOK34ju/OytXrPI4CHMIRnIAP51CFW6hBHSg8wDO8whtS6AW9o49Z6xLKZw7gj9DnD1ePjk4=</latexit><latexit sha1_base64="8Viwh2D00yJ5mcgJngUqqpm71Gs=">AAAB7XicbZDLSgMxFIZPvNZ6q7p0EyyCqzIjgm6EqhuXFewF2qFk0kwbm0mGJCOWoe/gxoUibn0fd76NaTsLbf0h8PGfc8g5f5gIbqznfaOl5ZXVtfXCRnFza3tnt7S33zAq1ZTVqRJKt0JimOCS1S23grUSzUgcCtYMhzeTevORacOVvLejhAUx6UsecUqssxpXT/gSh91S2at4U+FF8HMoQ65at/TV6SmaxkxaKogxbd9LbJARbTkVbFzspIYlhA5Jn7UdShIzE2TTbcf42Dk9HCntnrR46v6eyEhszCgOXWdM7MDM1ybmf7V2aqOLIOMySS2TdPZRlApsFZ6cjntcM2rFyAGhmrtdMR0QTah1ARVdCP78yYvQOK34ju/OytXrPI4CHMIRnIAP51CFW6hBHSg8wDO8whtS6AW9o49Z6xLKZw7gj9DnD1ePjk4=</latexit>

A�1<latexit sha1_base64="v78qxy3OlAHAI+5uT2vKGSPFNZ8=">AAAB7XicbZDLSgMxFIZPvNZ6q7p0EyyCG8uMCLqsunFZwV6gHUsmzbSxmWRIMkIZ+g5uXCji1vdx59uYtrPQ1h8CH/85h5zzh4ngxnreN1paXlldWy9sFDe3tnd2S3v7DaNSTVmdKqF0KySGCS5Z3XIrWCvRjMShYM1weDOpN5+YNlzJeztKWBCTvuQRp8Q6q3H1kJ36426p7FW8qfAi+DmUIVetW/rq9BRNYyYtFcSYtu8lNsiItpwKNi52UsMSQoekz9oOJYmZCbLptmN87JwejpR2T1o8dX9PZCQ2ZhSHrjMmdmDmaxPzv1o7tdFlkHGZpJZJOvsoSgW2Ck9Oxz2uGbVi5IBQzd2umA6IJtS6gIouBH/+5EVonFV8x3fn5ep1HkcBDuEITsCHC6jCLdSgDhQe4Rle4Q0p9ILe0cesdQnlMwfwR+jzB+Udjqs=</latexit><latexit sha1_base64="v78qxy3OlAHAI+5uT2vKGSPFNZ8=">AAAB7XicbZDLSgMxFIZPvNZ6q7p0EyyCG8uMCLqsunFZwV6gHUsmzbSxmWRIMkIZ+g5uXCji1vdx59uYtrPQ1h8CH/85h5zzh4ngxnreN1paXlldWy9sFDe3tnd2S3v7DaNSTVmdKqF0KySGCS5Z3XIrWCvRjMShYM1weDOpN5+YNlzJeztKWBCTvuQRp8Q6q3H1kJ36426p7FW8qfAi+DmUIVetW/rq9BRNYyYtFcSYtu8lNsiItpwKNi52UsMSQoekz9oOJYmZCbLptmN87JwejpR2T1o8dX9PZCQ2ZhSHrjMmdmDmaxPzv1o7tdFlkHGZpJZJOvsoSgW2Ck9Oxz2uGbVi5IBQzd2umA6IJtS6gIouBH/+5EVonFV8x3fn5ep1HkcBDuEITsCHC6jCLdSgDhQe4Rle4Q0p9ILe0cesdQnlMwfwR+jzB+Udjqs=</latexit><latexit sha1_base64="v78qxy3OlAHAI+5uT2vKGSPFNZ8=">AAAB7XicbZDLSgMxFIZPvNZ6q7p0EyyCG8uMCLqsunFZwV6gHUsmzbSxmWRIMkIZ+g5uXCji1vdx59uYtrPQ1h8CH/85h5zzh4ngxnreN1paXlldWy9sFDe3tnd2S3v7DaNSTVmdKqF0KySGCS5Z3XIrWCvRjMShYM1weDOpN5+YNlzJeztKWBCTvuQRp8Q6q3H1kJ36426p7FW8qfAi+DmUIVetW/rq9BRNYyYtFcSYtu8lNsiItpwKNi52UsMSQoekz9oOJYmZCbLptmN87JwejpR2T1o8dX9PZCQ2ZhSHrjMmdmDmaxPzv1o7tdFlkHGZpJZJOvsoSgW2Ck9Oxz2uGbVi5IBQzd2umA6IJtS6gIouBH/+5EVonFV8x3fn5ep1HkcBDuEITsCHC6jCLdSgDhQe4Rle4Q0p9ILe0cesdQnlMwfwR+jzB+Udjqs=</latexit><latexit sha1_base64="v78qxy3OlAHAI+5uT2vKGSPFNZ8=">AAAB7XicbZDLSgMxFIZPvNZ6q7p0EyyCG8uMCLqsunFZwV6gHUsmzbSxmWRIMkIZ+g5uXCji1vdx59uYtrPQ1h8CH/85h5zzh4ngxnreN1paXlldWy9sFDe3tnd2S3v7DaNSTVmdKqF0KySGCS5Z3XIrWCvRjMShYM1weDOpN5+YNlzJeztKWBCTvuQRp8Q6q3H1kJ36426p7FW8qfAi+DmUIVetW/rq9BRNYyYtFcSYtu8lNsiItpwKNi52UsMSQoekz9oOJYmZCbLptmN87JwejpR2T1o8dX9PZCQ2ZhSHrjMmdmDmaxPzv1o7tdFlkHGZpJZJOvsoSgW2Ck9Oxz2uGbVi5IBQzd2umA6IJtS6gIouBH/+5EVonFV8x3fn5ep1HkcBDuEITsCHC6jCLdSgDhQe4Rle4Q0p9ILe0cesdQnlMwfwR+jzB+Udjqs=</latexit>

A�1A = AA�1 = I<latexit sha1_base64="DDl91Soh9ai5q7t75HFDH3fWjJE=">AAAB/nicbZDLSsNAFIZP6q3WW1RcuRksghtLUgTdCK1udFfBXqCNZTKdtEMnkzAzEUoo+CpuXCji1udw59s4bbPQ1h8GPv5zDufM78ecKe0431ZuaXlldS2/XtjY3NresXf3GipKJKF1EvFItnysKGeC1jXTnLZiSXHoc9r0h9eTevORSsUica9HMfVC3BcsYARrY3Xtg+pDeuqOq+gSVWdo6LZrF52SMxVaBDeDImSqde2vTi8iSUiFJhwr1XadWHsplpoRTseFTqJojMkQ92nboMAhVV46PX+Mjo3TQ0EkzRMaTd3fEykOlRqFvukMsR6o+drE/K/WTnRw4aVMxImmgswWBQlHOkKTLFCPSUo0HxnARDJzKyIDLDHRJrGCCcGd//IiNMol1/DdWbFylcWRh0M4ghNw4RwqcAM1qAOBFJ7hFd6sJ+vFerc+Zq05K5vZhz+yPn8AZZWTLA==</latexit><latexit sha1_base64="DDl91Soh9ai5q7t75HFDH3fWjJE=">AAAB/nicbZDLSsNAFIZP6q3WW1RcuRksghtLUgTdCK1udFfBXqCNZTKdtEMnkzAzEUoo+CpuXCji1udw59s4bbPQ1h8GPv5zDufM78ecKe0431ZuaXlldS2/XtjY3NresXf3GipKJKF1EvFItnysKGeC1jXTnLZiSXHoc9r0h9eTevORSsUica9HMfVC3BcsYARrY3Xtg+pDeuqOq+gSVWdo6LZrF52SMxVaBDeDImSqde2vTi8iSUiFJhwr1XadWHsplpoRTseFTqJojMkQ92nboMAhVV46PX+Mjo3TQ0EkzRMaTd3fEykOlRqFvukMsR6o+drE/K/WTnRw4aVMxImmgswWBQlHOkKTLFCPSUo0HxnARDJzKyIDLDHRJrGCCcGd//IiNMol1/DdWbFylcWRh0M4ghNw4RwqcAM1qAOBFJ7hFd6sJ+vFerc+Zq05K5vZhz+yPn8AZZWTLA==</latexit><latexit sha1_base64="DDl91Soh9ai5q7t75HFDH3fWjJE=">AAAB/nicbZDLSsNAFIZP6q3WW1RcuRksghtLUgTdCK1udFfBXqCNZTKdtEMnkzAzEUoo+CpuXCji1udw59s4bbPQ1h8GPv5zDufM78ecKe0431ZuaXlldS2/XtjY3NresXf3GipKJKF1EvFItnysKGeC1jXTnLZiSXHoc9r0h9eTevORSsUica9HMfVC3BcsYARrY3Xtg+pDeuqOq+gSVWdo6LZrF52SMxVaBDeDImSqde2vTi8iSUiFJhwr1XadWHsplpoRTseFTqJojMkQ92nboMAhVV46PX+Mjo3TQ0EkzRMaTd3fEykOlRqFvukMsR6o+drE/K/WTnRw4aVMxImmgswWBQlHOkKTLFCPSUo0HxnARDJzKyIDLDHRJrGCCcGd//IiNMol1/DdWbFylcWRh0M4ghNw4RwqcAM1qAOBFJ7hFd6sJ+vFerc+Zq05K5vZhz+yPn8AZZWTLA==</latexit><latexit sha1_base64="DDl91Soh9ai5q7t75HFDH3fWjJE=">AAAB/nicbZDLSsNAFIZP6q3WW1RcuRksghtLUgTdCK1udFfBXqCNZTKdtEMnkzAzEUoo+CpuXCji1udw59s4bbPQ1h8GPv5zDufM78ecKe0431ZuaXlldS2/XtjY3NresXf3GipKJKF1EvFItnysKGeC1jXTnLZiSXHoc9r0h9eTevORSsUica9HMfVC3BcsYARrY3Xtg+pDeuqOq+gSVWdo6LZrF52SMxVaBDeDImSqde2vTi8iSUiFJhwr1XadWHsplpoRTseFTqJojMkQ92nboMAhVV46PX+Mjo3TQ0EkzRMaTd3fEykOlRqFvukMsR6o+drE/K/WTnRw4aVMxImmgswWBQlHOkKTLFCPSUo0HxnARDJzKyIDLDHRJrGCCcGd//IiNMol1/DdWbFylcWRh0M4ghNw4RwqcAM1qAOBFJ7hFd6sJ+vFerc+Zq05K5vZhz+yPn8AZZWTLA==</latexit>- There exists a square matrix, , such that

Page 146: COMP 414/514: Optimization - Algorithms, Complexity and

Conclusion

- We have set up background and notation w.r.t. linear algebra

- We saw a toy example where non-convex operations happen

Page 147: COMP 414/514: Optimization - Algorithms, Complexity and

Conclusion

- We have set up background and notation w.r.t. linear algebra

Next lecture

- Brief introduction to convex optimization and related topics

- We saw a toy example where non-convex operations happen

Page 148: COMP 414/514: Optimization - Algorithms, Complexity and

Demo