Math 2270 - Assignment 12

Dylan Zwick

Fall 2012

Section 6.5 - 2, 3, 7, 11, 16Section 6.6 - 3, 9, 10, 12, 13

Section 6.7-1,4,6,7,9

1

6.5 - Positive Definite Matrices

6.5.2 - Which of A1,A2,A3,A4 has two positive eigenvalues? Use the test,don’t compue the )‘s. Find an x so that xTA1x < 0, so A1 fails thetest.

10 \Ai=(?) A2=(j

)A3=(’

100)10 \

A4=(1 o)

70

5

___

n I/

5 40 5 / ( !fl I

43 7

€4i

I 7 O Iy/o/- /a/6//

r e i

5 L (f*6L +7i)4Y1

7

*2

5 +(L)

(I) 0/k)•

H

c’.0)I.

oc)

0)cj:

II

U 0)

cE0)

II0)U

.‘-Th

-0)U

—.

0r•I

Th

—

___

0i:

cN

ciII

c-)0

:

c.

—

CJD

(S

6.5.11 - Compute the three upper left determinants of A to establish thepositive definiteness. Verify that their ratios give the second andthird pivots.

/2 2 0A= ( 2 5 3

\\0 3 8

L-

5 Z393 —

O7

L (o-i) -L(1) Z(I5)3O

A P

.z / 7J

Q3/

9

r{ 7 / 5(o4 q/j

h ir 5

6.5.16 - A positive definite matrix cannot have a zero (or even worse, anegative number) on its diagonal. Show that this matrix fails to havexTAx> 0:

/4 1 1 /x’

( x x2 X3 ) ( 1 0 2 f x2 ) is not positive when2 5) \X3)

(x1,x2,x3)= ( —

C( 7Iy3

- l9y3

ict 19

H -i 7)4 1/ / 3

I 0 / (lt)

(o 170)(Y

6.6 - Similar Matrices

6.6.3 - Show that A and B are similar by finding M so that B M’AM:

A=( 3) and B=( 8)A=( ) and B=(1 ‘)A=( and B=( ).

10)/I(:) ( )7 6 (it)

M(Id)

I /10- i } ) (1 ) 4/ 4cjJ

(U(oii/(jq (d

7

Ioj

iN

1)C)

Ccj

j

C(D

UC

CCr

)0

IIIi

-- 0

If

c(N

II

ç0

6.6.10 - By direct multiplication, find J2 and J3 when

=( ).Guess the form of jk• Set k = 0 to find J°. Set k = —ito find J’

A

A I / Z/ (AY AL1( UA) // A)

T )k (. 1 ic-j

io

I

9

6.6.12 - These Jordan matrices have eigenvalues 0, 0. 0. 0. They have twoeigenvectors (one from each block). But the block sizes don’t matchand they are not similar:

o 1 0 oJ—

0 0 0 0— 00011

0 0 0 o)

/o 1 0 oK—

0 0 1 0

0 0 0 0)— 0 0 0 0

For any matrix M, compare JM with MK. If they are equal showthat M is not invertible. Then M’JM = K is impossible: J is notsimilar to K.

and

I / ThL 3 1 L/

M , / o

I I LI 1 L q IRH IflLI

o o /,I )Ti

/k / o

0 10 I 3I 0Q

e (c/C(7

10 oF M+Jr fof f/M)

6.6.13 Based on Problem 12, what are the five Jordan forms when ) =

0,0,0,0?

/ OO

/ o ooJ / 0000

7/

7Ooo/ QQ

01

/ /(00

( oo0O)

0000

11

6.7 - Singular Value Decomposition (SVD)

6.7.1 - Find the eigenvalues and unit eigenvectors v1,v2 of ATA. Then findu1 = Av1/u1:

Verify that u1U,,v.

is a unit eigenvectors of AAT. Complete the matrices

SVD

5oA

//o

Lo

A=( and ATA(10 20

20 40and AAT =

(515

1545

/1 2N3 5) = ( u1

c1d-

(0-

/u ON0

v2)T

ij0)t1)/0)-fO -

A

12

c-

3

r’J

—‘

11I.’

(r\\

011

-

0—

6.7.4 - Find the eigenvalues and unit eigenvectors of ATA and AAT. Keepeach Av = ou:

Fibonacci Matrix A=( )

1 -3jHAl

A

Construct the singular value decomposition and verify that A equalsUZVT.

AA AA

1 -

L

L

)V

13

cç

ci I’)

11

9

H N

><

-1-

\

Th‘I

—

—

H

Ii

‘H

6.7.6 - Compute ATA and AAT and their eigenvalues and unit eigenvectors for this A:

Rectangular Matrix A= ( ).

Multiply the matrices UVT to recover A. has the same shape as

411 o

I 1-A

(I -A) Th42(/A)o / HA

(- /3/(/-A)5JA

0, ,3

14

ci

__

__

__

__

__

__

__

__

__

__

_

——

__

_

——

—

——

—

______

_______

—f’

)_-

—

r—

.-

ciH

f

‘Ici

ci-cc

11

I\1

N

___

II

—

/1 .1

4

t, a Iei oJr *i3 / 1/r%

I3 ôeY 4a exi3J

0 0’

to

6.7.7 - What is the closest rank-one approximation to that 3 by 2 matrix?

5Q)J

o[l

1

j

L

7-

15

6.7.9 - Suppose u1,. . . u and v1,. . . , vt-, are orthonormal bases for W.Construct the matrix A that transforms each v3 into u3 to give Av1 =

(1

-

(I

7

vT

/

T

U

C-- IA

LJ\

16