9 Show that the characteristic polynomials of two similar matrices are equal.

10 Show that U Y U Y STSRQ

hfg¢edbA `aH ¢ §WU

¦c XV

for any matrix norm. [Hint: Use the Jordan canonical form.]

YY Y ¢ Y pY ¢ Y

i

11 Let be a nonsingular matrix and, for any matrix norm , de¬ne . Show

B BY

5 Y ¢qB

that this is indeed a matrix norm. Is this matrix norm consistent? Show the same for and

Y ¢ rsY r

where is also a nonsingular matrix. These norms are not, in general, associated with

B

any vector norms, i.e., they can™t be de¬ned by a formula of the form (1.7). Why? What about

Y I¤¢uB Y hY ¢ Y

t

the particular case ?

HF B

12 Find the ¬eld of values of the matrix

&

"

%¢

v"

"

and verify that it is not equal to the convex hull of its eigenvalues.

uY u d¥ £ ¡ ¨ ¢¤¥

n ¦¢¥

¤¤§ ¥ §¢

¡

£ ¡© ¥ ¡ © ¥ ¡ ¥ © ¥ ¥

¥ ¨

13 Show that for a skew-Hermitian matrix ,

d§¦ ¦¤c £¡

" f¥ ¥ ¢ for any 5 ¨©¥

14 Given an arbitrary matrix , show that if for all in , then it is true that

"% d¥I¥¤c

f¦

¥

5 &%$¦ #" 9§¦ ¤ f ¦ ¦¤c

¨ ! " f c

[Hint: Expand .]

f 10hf )('c ¤c

¦

15 Using the results of the previous two problems, show that if is real for all in , f §¦ 2¢ec

¥¥ ¥ 3

then must be Hermitian. Would this result be true if the assumption were to be replaced by:

¢

is real for all real ? Explain.

fd¥§I4ec

¦¥¢ ¥

16 The de¬nition of a positive de¬nite matrix is that be real and positive for all real vectors dI4ec

f ¥¦ ¥ ¢

. Show that this is equivalent to requiring that the Hermitian part of , namely, , ¤g¢ ec H5

¢ ¢

f

¥

be (Hermitian) positive de¬nite.

£ 8 5 ¢

17 Let and where is a Hermitian matrix and is a Hermitian Positive

7H F £ H ¢ £

6 6 6

5¢ 5¢

De¬nite matrix. Are and Hermitian in general? Show that and are Hermitian

H¢ H¢

(self-adjoint) with respect to the -inner product. £

18 Let a matrix be such that where is a polynomial. Show that is normal. [Hint:

¢ g¢ec $ ¢

f $ ¢

Use Lemma 1.2.]

19 Show that is normal iff its Hermitian and skew-Hermitian parts, as de¬ned in Section 1.11,

¢

commute.

20 Let be a Hermitian matrix and a Hermitian Positive De¬nite matrix de¬ning a -inner

¢ £ £

product. Show that is Hermitian (self-adjoint) with respect to the -inner product if and only

¢ £

if and commute. What condition must satisfy for the same condition to hold in the more

¢ £ £

general case where is not Hermitian? ¢

21 Let be a real symmetric matrix and an eigenvalue of . Show that if is an eigenvector

¢ ¡ ¢ 9

associated with , then so is . As a result, prove that for any eigenvalue of a real symmetric

¡ @9

matrix, there is an associated eigenvector which is real.

G0 ¨¦ ¦ ( ¦ & ¥PQ¦ %IC EFDB

&

22 Show that a Hessenberg matrix such that , cannot be deroga-

" H GH C

A W5W5

5

tory.

23 Prove all the properties listed in Proposition 1.6.

& W9 H F ¢ US

24 Let be an -matrix and two nonnegative vectors such that . Show that

¢0 ¦ 9

R S T

V

is an -matrix.

T X9 ¢

S R

25 Show that if then . Conclude that under the same assumption,

£AY 8¢A`aY Y Y £ bA8IT(¢ aY

T£` ¢

` `

5 £Y ` 5 ¢

we have .

5 d¦ c

26 Show that for two orthogonal bases of the same subspace of we have u¥ H c H c

cH R

#eI¥ ©c 5 c

¥ ! ¦ 5

.

27 What are the eigenvalues of a projector? What about its eigenvectors?

5f 5 f a!f

28 Show that if two projectors and commute, then their product is a projector.

Hf

Hf

What are the range and kernel of ? f

d¥ © g¢ ¤© ©

n¥ u¡ § £ ¥

© ©

¡

29 Consider the matrix of size and the vector ,

¢

¨¥

04&4v&

0 &0

& 99 8 0

&

34 &

344 99 8

4 99 0 W5W5

&5

040 & "

&

& ( ¢£&

44 44 99