Lecture 39: Quick review from previous lecture

• LetA 2 Mm⇥n (m⇥n real matrices) of rank r. SupposeA has positive singular

values �1 � . . . � �r and corresponding right singular vectors v1, . . . ,vr and

left singular vectors u1, . . . ,ur. Then

A = �1u1vT1 + . . . + �rurv

Tr .

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Today we will

• review some concepts

- Lecture will be recorded -

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• Exam 3: 12/16 (Wednesday) in lecture.

• Practice Exam is on Canvas now.

• Additional o�ce hours next Tuesday from 10 am-11:50 am.

MATH 4242-Week 15-1 1 Fall 2020

-O

- -

= " ' --

I' l" I Tired;:S,

( # 9 solution is modified)

Problem 12: Find a 2-by-2 matrix A with eigenvalues 2 and �3 and corre-

sponding eigenvectors (1,�1)Tand (1, 0)T .

Problem 13: Find a 2-by-3 matrix having rank 1 whose singular value is 2, left

singular vector is (1, 2)T/p5, and right singular vector is (1, 0, 1)T/

p2.

[We have discussed in Lecture 38.]

MATH 4242-Week 15-1 2 Fall 2020

-

I, E

.

A [ v, Va ) = ( 2v, -34 )= [ v, 47 f! go ]

A = Cv. v. If ! 1) fu, rift= C - so I

- #

Problem 14: Write out the SVD of the matrix A =

✓1 �1

1 �1

◆.

MATH 4242-Week 15-1 3 Fall 2020

ATA = (I-

I ) has eigenvalues 4 , O

singular value ,T1¥ = ATA - 41 = (II ) .

v. = (tyrant= ATA = (II) . " = ( th → Kera

.

I. :

Kern cokerAwut

u,= Aft = l !)g, → ing A

Find orthonormal basis for askers :

uz t u , ⇒ u- = (th.

A- Cu . a.) 17151 III. ] (Tulisa)= f. u . ] 12 ] I v.

T ) (Reduced

= f " hi Cyr.-ya ,

suo)-

A- = 2 u , v.T

Problem 15: Suppose A is a 2-by-2 symmetric matrix with eigenvalues 3 and

�4. Find the operator norm of A and the Frobenius norm of A.

Problem 16: Suppose A has characteristic polynomial pA(�) = �2 � 2� + 2.

Find the determinant of A and the trace of A.

Problem 17: Suppose A has characteristic polynomial pA(�) = �2 � 2� + 2.

Find the characteristic polynomial of A�1.

MATH 4242-Week 15-1 4 Fall 2020

( T 4,02=3 )-O

enable)- Tnatuenatnx norm !

HAIL = max { I dial,la . I / = 4

HAH#

= Fit = 5. #

A has eigenvalues a, b .Ps (A) = D

'- za +2 -

detA=x ,- -- an

I.' 'II' c'a''I.7. + as far.net.in .

doc A- = ab =③⇒ doestrot) = atb = 2

. #

deeks) det ( A" - az )=doelA④#

=A_ der ( A" ( I - AA ) )= dot ( RA" ( II - Al )( doe l -as' ca - in) ¥157,= diet (ILA

-t ) det ( A -⑤z )± -45 @etA'' ) (Hi - 4kHz)

-742= 172 - R t Yz

.

Problem 18: Suppose A = ATis a symmetric 2-by-2 matrix, and detA = 6.

Suppose that Av = 2v, where v = (1, 1)T . Write the spectral factorization of A.

Problem 19: Let V = P (1)be the space of polynomials of degree 1, andW =

P (2)be the space of polynomials of degree 2. Let L[p](x) =

R x0 p(t)dt denote the

integration operator. Find the matrix representation of L in the monomial bases

of V and W .

MATH 4242-Week 15-1 5 Fall 2020

-

D , .V,= like A = QDQT -- QDQT

⇐ dot A = 2 . Az ⇒ a, =3where Iis orthogonal

since A-AT,ht v

,⇒ V.= ( IKE

.

A -- %.

13519% .

'' '

*→orthogonal

Li p '' '→ plz )-1X

, if I x? x, i)

( ( att b ) = a L Cx ) t BLE 17= [ LEXI Lin ) ( f )

• Lfx ) = f! it dt = II → (kg )• LCM = f! I dt = ×

→

fog )Lll :) ] -- f ! Ill :)

Problem 20: Suppose A is a 3-by-3 matrix with singular values 1,2, and 3.

What is the condition number of A? What are the singular values of A�1? What

are the singular values of AT? What is the determinant of A?

Problem 21: Suppose A is a 3-by-3 symmetric matrix of the form

A = 2u1uT1 � 3u2u

T2 + 9u3u

T3 ,

where u1,u2,u3 2 Rnare nonzero column vector and are orthonormal. What is

the condition number of A? What are the singular values of A�1? What are the

singular values of AT? What is the determinant of A?

MATH 4242-Week 15-1 6 Fall 2020

9 On T,

- -

1) KIA) =%, = 3

→ x. x. x. . then!::II÷,3) I,2, 3 = It 6 It

41 If =E6_

-(spectral factorized

=Cu , um,] (Z-3 g) fu.u.u.TT•

T, = 9 , 02=3 ,03=2 (Ti = lait

t) KCA ) = 9/2=4.5 since A- AT)2) Ya

,43

,42

3)9 , 3,2

4) dot A = 2C -3) g = -54-#

Problem 22: Suppose A is a matrix with singular values 2, 3 and 8. Suppose

u and v are the left and right singular vectors of A with singular value 8, and let

B = 8uvT. Find kA� Bk2 and kA� BkF .

Problem 23: SupposeA = 2uvT, where u = (1,�1)

T/p2 and v = (1, 1)T/

p2.

Let b = (1, 0)T . Find all least squares solutions to Ax = b. That is, find all

vectors x that minimize kAx�bk2. Also, find the unique vector x that minimizes

kAx� bk2 and has the smallest Euclidean norm.

MATH 4242-Week 15-1 7 Fall 2020

A- = 8 UVT t 3 U.zv.tt 2 Us VstB = SEUVT : Best rankle approximation ofA .

[ = 8 UVTT 3 cravat : Best rank 2 s

-

A -B = 3 Uz VT t 2434T has singularvalues 3

,2

.

HA - 13112 = 3 ; HA -BH,-_TE

O=D . #

-

-

-" A = [ u ] 127 CUT (Reduced

At = C v ) ( Yz ] HI

xx = Atb = ( YT.).

4 Ft Ikerd : 2-= (t, )t ,orthogonal-

=fY¥ ) t (f) t , te EIR .

Problem 24: Find the general solutions x(t) = (x1(t), x2(t), x3(t))T to the

following system of di↵erential equations:

8<

:

x01 = 4x1 + x3x02 = 2x1 + 3x2 + 2x3x03 = x1 + 4x3

.

Here xi = xi(t) is a di↵erentiable function of variable t and x0i(t) =ddtxi(t), i =

1, 2, 3. Simplify your answer.

MATH 4242-Week 15-1 8 Fall 2020

A- ( & §! ) ,D= 3

,

3,5

.

⇒ .

v. = ( Y) , h= (g). v. = ( fl

A-- V D V"

nth D= ( { II )✓ = fu, v. is]

Then X'

= Ax= VDV

-"x

⇒ ( y )'

= Dy d key -_V'x

⇒ s=l÷÷÷,Then x=vy= (

"' e:e¥EF¥c.e't*GET . #