Lecture 39: Quick review from previous lecture

• We have reviewed permuted LU factorization, inverse of a matrix, positive(semi)definite, determinant, solving a linear system and many others.

—————————————————————————————————Today we will review some concepts.

- Lecture will be recorded -

—————————————————————————————————

• Information for Final Exam and Course Grade has been posted on Canvas, see”Announcements”.

MATH 4242-Week 16-1 1 Spring 2020

Problem 14: Find the QR factorization of A =

0

@0 �1 00 0 �32 �1 �3

1

A. Clearly

identify the orthogonal matrix Q and the upper triangular matrix R.

Problem 15:

a) Find the matrix norm of A =

0

@�4 0 00 �1 00 0 7

1

A, with respect to the standard

Euclidean norm kyk2 =py21 + y22 + y23 on R3.

MATH 4242-Week 16-1 3 Spring 2020

Ant 492 493

V. = a , = ( E ) , g ( Y)v. = ar - 9Y' 4=1 :o) . E- ( Ilvs = as - c%Y u

.- ca÷iu= 1%1,95. II)

Then a -

- f ! ! ÷ !=

* l ! "

÷ ) . "it! .is=p:

"

: -41. #

NAH = Fia; tail = 7.

b) Suppose kxk = 3 and kyk = 1. What is the maximum possible value for hx,yi?What relationship must hold between x and y if this value is to be achieved?

c) Suppose hx,yi = 0, kxk = 2, kyk = 1, and z = 2x � 3y. What is hx, zi?What is hy, zi? What is kzk?

Problem 16: Find all vectors in R3 orthogonal to both (1, 2, 0)T and (2, 5, 2)T .

MATH 4242-Week 16-1 4 Spring 2020

( x. x > = 11×112 .

1) / Cx, y> / E 11×1111911.=3

.

maximum valueis 3

'

#2) X =3 Y-

'

#

- o

1) (X. Z > = ( X, 2×-3 y) = (X , 2x> - CX , 377=24×7 -34%= 211×112 = 8

2) Cy,z> = - 3

.

-#

3) 112112 = Cz,z > = CTx-tyx-fip-44.is> - 647%

>O

- 6#t gey.D= 4.22+9=25 .

- - A - ""

VIT X = o - Y K.

viii. o' ( LEE ) x = O .

A①

( o' ? ;) .

I = ( Y) z .

e-R.

Problem 17: Write down the 2-by-2 matrix A satisfying Av1 = w1 andAv2 = w2, where v1 = (1, 1)T , v2 = (�1, 1)T , w1 = (1, 1)T , and w2 = (�2,�2)T .

Problem18: Find a 2-by-2 matrix A with eigenvalues 2 and �3 and correspond-ing eigenvectors (1,�1)T and (1, 0)T .

Problem 19: Write out the SVD of the matrix A =

✓1 �11 �1

◆.

MATH 4242-Week 16-1 5 Spring 2020

JV, -_ Wi

Av, =wz: A sk !

-

- Cw , wa ]

A = Ew, wz ) s -' = If } I ).

#

't.

"vz

AV, = 2 V,

Av. =-3 v,i Alj]=l4 Va ) f} µA- stools - '

= ( III.*

STS = ( I ? ).

o= dee CATA - AI )a = 4

,O

.

=

¥4. (STA- GI) r -

- o, v = EH !

u= A¥= ¥41.

Reduced SD of Ais A- rill) CDC# E).

*.

Problem 20: Find a 2-by-3 matrixA having rank 1 whose singular value is 2, leftsingular vector is u = (1, 2)T/

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

p2,

that is, Av = 2u.

Problem 21: Suppose A is a 2-by-2 real matrix for which 1�2i is an eigenvalue.Find the trace and determinant of A.

Problem 22: 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.

MATH 4242-Week 16-1 6 Spring 2020

=

A- = Luv 'T.

= Iff) He 0¥= rs

=U[2) VT = ⇐ ( f ) ( I 01)(Reduced SVD)

, o l= ( z o z ). #

I ¥2 i are eigenvalues of A .

det A = D , -- . An = (Hzi ) ( I- zi ) =I#

Trace of A = a" t . - - t Ann = ( Aij )#n

.= D, t Az

= Ctu ) + ( I- zi) =L . #

Operator nom of A : HAH -

- qq.nl " il = IFrobenius norm of A : Il Atf = It

= 5- #

.

Problem 23: Suppose A has characteristic polynomial pA(�) = �2 � 2� + 2.Find the determinant of A.

Problem 24: Suppose A has characteristic polynomial pA(�) = �2 � 2� + 2.Find the characteristic polynomial of A�1.

Problem 25: Suppose A = AT is a symmetric 2-by-2 matrix, and detA = 6.Suppose that Av = 2v, where v = (1, 1)T . Write an spectral factorization of A.

MATH 4242-Week 16-1 7 Spring 2020

→def A = 17.172=2

. # /v

Pala ) = (R - D . ) ( R - da ) = - - - - t

"doe CA-92)

TE

Pa. . (5) = det ( A " - II )=det( At ( I - IA ) )= det A" dee ( I - E.Al.= detail

"

. det (C- A) (A - ¥2) )deter 't

= doe EAT detCA-EIPs.tl-- Ies -- I

= 'T E k¥5 - ¥+2) = I'- Itt.#

6 -_ dots = 2 - a ⇒ 17=-3.Vz: Eigenvector of 17=3

,

Va is orthogonal to V=

Taking Vz± (t ).

A-- c ' toil'

*

Problem 26: 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 basesof V and W .

Problem 27: 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? Whatare the singular values of AT? What is the determinant of A?

MATH 4242-Week 16-1 8 Spring 2020

P'' '= ( x. I 1 ,

P'''= I *2. x. it

( (x ) = Jot tdt = I X ' t O - X -10.1

L ( 17 = f! Idt = O X't X x o - I

Thus, matrix representation of L is

A- ( ÷.It

. #

→

1) KIA ) = 3,1 =3

4 Sir , of A"

= I. I

'

,'s

3) su, of AT = I,

2,3.

4) dot A = det (UI VT ) = £6( full sup) .

#

TMZ that UTU =UUT= II = det UT dei U = (docu )

'

det U =L ! Jsimilarly , der V = It

.

Problem 28: Suppose A is a matrix with singular values 2, 3 and 8. Supposeu and v are the left and right singular vectors of A with singular value 8, and letB = 8uvT . Find kA� Bk2 and kA� BkF .

Problem 29: Suppose A = uvT , 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 allvectors x that minimize kAx�bk2. Also, find the unique vector x that minimizeskAx� bk2 and has the smallest Euclidean norm.

MATH 4242-Week 16-1 9 Spring 2020

Eu an matrix norm

I = f UVT t 3 Unit 2 Us.

A -B = Susu'T t 2434T .

So, HA-Blk = 3

11 A- BHf= ME =D

.

' l At -_ v." Cnut..= ( ¥

.)

E- At b = ( II ).

.

has the smallest Euclideannorm.

4 All least squaressolution are X*t E

,

where ZE Ker A .

Finding a basis for Ker A :

we knew V=rT( !) is in wings, soE- (4)t is in Ker A .

Then *tz= (E) t.tt ) , THR . #