Upload
jacqui
View
45
Download
3
Tags:
Embed Size (px)
DESCRIPTION
Modified Gary Larson Far Side cartoon. Matlab. Linear Algebra Review. Matrices can represent sets of equations!. a 11 x 1 +a 12 x 2 +…+ a 1n x n =b 1 a 21 x 1 +a 22 x 2 +…+a 2n x n =b 2 … a m1 x 1 + a m2 x 2 +…+ a mn x n = b m What’s the matrix representation?. Vectors. Matrices. - PowerPoint PPT Presentation
Citation preview
Modified Gary Larson Far Side cartoon
Matlab
Linear Algebra Review
Matrices can represent sets of equations!
a11x1+a12x2+…+a1nxn=b1
a21x1+a22x2+…+a2nxn=b2
…
am1x1+am2x2+…+amnxn=bm
What’s the matrix representation?
€
A =
a11 a12 ... a1na21 a22 ... a2n
...am1 am2 ... amn
⎡
⎣
⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥
€
x =
x1x2.xn
⎡
⎣
⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥
€
b =
b1b2.bm
⎡
⎣
⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥
€
Ax = b
Vectors
Matrices
U = [umn] =
u11 u12 … u1n
u21 u22 … u2n
…um1 um2 …
umn
The general matrix consists of m rows and n columns. It is also known as an m x n (read m by n) array.
Each individual number, uij, of the array is called the element
Elements uij where i=j is called the principal diagonal
Transpose of a Matrix
Matrix & Vector Addition
Vector/Matrix addition is associative and commutative(A + B) + C = A + (B + C); A + B = B + A
Matrix and Vector Subtraction
Vector/Matrix subtraction is also associative and commutative(A - B) - C = A - (B - C); A - B = B - A
Matrix and Vector Scaling
X
ax
• For addition and subtraction, the size of the matrices must be the same
Amn + Bmn = Cmn
• For scalar multiplication, the size of Amn does not matter
• All three of these operations do not differ from their ordinary number counterparts
• The operators work element-by-element through the array, aij+bij=cij
Vector Multiplication
• The inner product or dot product
v
wa
€
v ⋅w = (x1,x2) ⋅(y1,y2) = x1y1 + x2y2
The inner product of vector multiplicationis a SCALAR
€
v ⋅w = (x1,x2) ⋅(y1,y2) =||v ||⋅ ||w || cosα
Inner product represents a row matrix multiplied by a column matrix. A row matrix can be multiplied by a column matrix, in that order, only if they each have
the same number of elements!In MATLAB, in order to properly calculate
the dot product of two vectors use
>>sum(a.*b)
element by element multiplication (.*)sum the results
A . prior to the * or / indicatesthat matlab should perform the array or element by element calculation rather
than linear algebra equivalent
or
>>a’*b
• The outer product
A column vector multiplied by a row vector.
In Matlab:
>>a*b’
ans =
24 6 30 8 2 10 -12 -3 -15
The outer product of vector multiplicationis a MATRIX
Two matrices can be multiplied together if and only if
the number of columns in the first equals the number of rows in the second.
Matrix Multiplication
pmmnpn BAC
m
kkjikij bac
1
The inner numbers have to match
a21 a22
In MATLAB, the * symbol represents matrix multiplication :
>>A=B*C
B C A
C*B #taken from previous slide ans =
19 18 17
16 24 32
32 24 16
nnnnnnnn ABBA
• Matrix multiplication is not commutative!
• Matrix multiplication is distributive and associative
A(B+C) = AB + BC
(AB)C = A(BC)
Revisit the vector example
>>a'*b
ans =
11
a1x3 * b3x1 = c1x1
>>a*b’
ans =
24 6 30 8 2 10 -12 -3 -15
a3x1 * b1x3 = c3x3
Dot products in Matlab(using this form – built in functions - don’t have to match dimensions of vectors – can mix
column and row vectors – although they have to be the same length)
>> a=[1 2 3];>> b=[4 5 6];
>> c=dot(a,b)c = 32
>> d=dot(a,b’)d = 32
Dot products using built-in functionFor matrices – does dot product of columns.
Essentially treating the columns like vectors.The matrices have to be the same size.
>> a=[1 2;3 4]a = 1 2 3 4
>> b=[5 6;7 8]b = 5 6 7 8
>> dot(a,b)ans = 26 44
Determinant of a Matrix
or follow the diagonals
det (A) = a11a22a33+ a12a23a31 + a13a21a32 − a11a23a32 − a12a21a33 − a13a22a31
Cross-product of two vectorsThe cross product a × b is defined as a vector c that is
perpendicular to both a and b, with a direction given by the right-hand rule and a magnitude equal to the area of the parallelogram that the vectors span.
€
a × b = a b sinθ
c = a2b3-a3b2 = 2*5 – (-3)*1 = 13
a3b2-a1b3 -3*4 – 6*5 -42
a1b2-a2b1 6*1 – 2*4 -2
Cross products in Matlab(using this form – built in functions - don’t have to match dimensions of vectors – can mix
column and row vectors – although they have to be the same length)
>> a=[1 2 3];>> b=[4 5 6];
>> e=cross(a,b)e = -3 6 -3
>> f=cross(a,b’)f = -3 6 -3
>> g=cross(b,a)g = 3 -6 3
For matrix – does cross product of columns. (one of the dimensions has to be 3 and takes other dimension as additional vectors)
>> a=[1 2;3 4;5 6]a = 1 2 3 4 5 6
>> b=[7 8;9 10;11 12]b = 7 8 9 10 11 12
>> cross(a,b)ans = -12 -12 24 24 -12 -12
Matrix Operators
• + Addition• - Subtraction• * Multiplication• / Division• \ Left division • ^ Power• ' Complex conjugate transpose• ( ) Specify evaluation order
Array Operators
• + Addition• - Subtraction• .* Element-by-element multiplication• ./ Element-by-element division.
• A./B: divides A by B by element
• .\ Element-by-element left division• A.\B divides B by A by element
• .^ Element-by-element power• .' Unconjugated array transpose
• the signs of imaginary numbers are not changed, unlike a regular matrix transpose
Operators as built-in commands
plus - Plus + uplus - Unary plus + minus - Minus - uminus - Unary minus - mtimes - Matrix multiply * times - Array multiply .* mpower - Matrix power ^ power - Array power .^ mldivide - Backslash or left matrix divide \ mrdivide - Slash or right matrix divide / ldivide - Left array divide .\ rdivide - Right array divide ./ cross - cross product
Multiplication in Matlab
>> x=[1 2];>> y=[3 4];
>> z=x*y’z = 11
>> w=x.*yw = 3 8
>> z=x'*yz = 3 4 6 8
Regular matrix multiplication – in this case with vectors 1x2 * 2x1 = 1x1 => dot productElement by element multiplication
Regular matrix multiplication – in this case with vectors 2x1 * 1x2 = 2x2 matrix
Division in MatlabIn ordinary math, division (a/b) can be thought of as a*1/b or a*b-1.
A unique inverse matrix of B, B-1, only potentially exists if B is square. And matrix multiplication is not communicative, unlike ordinary multiplication.
There really is no such thing as matrix division in any simple sense.
/ : B/A is roughly the same as B*inv(A). A and B must have the same number of columns for right division.
\ : If A is a square matrix, A\B is roughly the same as inv(A)*B, except it is computed in a different way. A and B must have the same number of rows for left division.
If A is an m-by-n matrix (not square) and B is a matrix of m rows, AX=B is solved by least squares.
The / and \ are related
B/A = (A'\B')’
Inverse of a Matrix
the determinant
the principal diagonal elements switch
the off diagonal elements change sign
• Square matrices with inverses are said to be nonsingular
• Not all square matrices have an inverse. These are said to be singular.
• Square matrices with determinants = 0 are also singular.
• Rectangular matrices are always singular.
Right- and Left- Inverse
If a matrix G exists such that GA = I, than G is a left-inverse of A
If a matrix H exists such that AH = I, than H is a right-inverse of A
Rectangular matrices may have right- or left- inverses, but they are still singular.
Some Special Matrices
• Square matrix: m (# rows) = n (# columns)
• Symmetric matrix: subset of square matrices where AT = A
• Diagonal matrix: subset of square matrices where elements off the principal diagonal are zero, aij = 0 if i ≠ j
• Identity or unit matrix: special diagonal matrix where all principal diagonal elements are 1
Linear Dependence
2a + 1b = c
a b c
Linear Independence
There is no simple, linear equation that can make these vectors related.
a b c
Rank of a matrix
Acknowledgement
• This lecture borrows heavily from online lectures/ppt files posted by
• David Jacobs at Univ. of Maryland• Tim Marks at UCSD• Joseph Bradley at Carnegie Mellon