”’

uCY l 0

)

£

The most important cases are again those associated with . The case

x

¡

) is of particular interest and the associated norm is simply denoted by and

¡

# #

called a “¡ -norm.” A fundamental property of a -norm is that ¡

# ˜ # # # ( ˜

an immediate consequence of the de¬nition (1.7). Matrix norms that satisfy the above

property are sometimes called consistent. A result of consistency is that for any square

matrix ,

' '

#

# #

'

In particular the matrix converges to zero if any of its -norms is less than 1. ¡

The Frobenius norm of a matrix is de¬ned by

6 0 8

3

H…„‚ 1‘0 ( & 1(” &… 5 6 t

1§ ¨ 2 ¦§¥

A

0

X

‘

This can be viewed as the 2-norm of the column (or row) vector in consisting of all the 4

3

˜

columns (respectively rows) of listed from to (respectively to .) It can be shown

d¥ ¥ z© §¥ ¥ ¡ p ¥ ¥

n e ¨ e ©¢

¡£ © ¨¤© ¦

¥¡ ¡

that this norm is also consistent, in spite of the fact that it is not induced by a pair of vector

norms, i.e., it is not derived from a formula of the form (1.7); see Exercise 5. However, it

does not satisfy some of the other properties of the -norms. For example, the Frobenius ¡

norm of the identity matrix is not equal to one. To avoid these dif¬culties, we will only use

the term matrix norm for a norm that is induced by two norms as in the de¬nition (1.7).

Thus, we will not consider the Frobenius norm to be a proper matrix norm, according to

our conventions, even though it is consistent.

The following equalities satis¬ed by the matrix norms de¬ned above lead to alternative

de¬nitions that are often easier to work with:

H… ‚ 1‘ ( & §¦¥1¢0 ( … 0

¤ t

¨ ¢ §¥

¦

%%%%"

"$$$

0 ”

H… ‚ 0 (” … ¦ ¤ 1(

& §¥¢ t

¡ ¨ ¤¦ §¥

£¦

%%%%"

"$$$ 0

6 0 ‘

6 0

6 0 C I X B©¡ 6 0 C I B ¡ 6

t

©¦ ¦ §¥

¨¦

¨ ¨

C I ( B C I B t

1© 1 1©

¨ & ¦ §¥

¦

¨ ¨

i)11Y ˜P DIP

As will be shown later, the eigenvalues of are nonnegative. Their square roots

are called singular values of and are denoted by . Thus, the relation X

6 0

(1.11) states that is equal to , the largest singular value of . X

¥ „ B ¡

U ©¨¦

T § ¥

From the relation (1.11), it is clear that the spectral radius is equal

C

to the 2-norm of a matrix when the matrix is Hermitian. However, it is not a matrix norm

in general. For example, the ¬rst property of norms is not satis¬ed, since for

7

7 7

C B ¡

a

7 7

we have while . Also, the triangle inequality is not satis¬ed for the pair ,

G ˜

and where is de¬ned above. Indeed,

C ˜™— B ¡ p C ˜B ¡ — C B ¡

7

while