’ B (…

&

@

¢ ¢

3

'

&

C

' ¤ £

Since , each of the terms in this ¬nite sum converges to zero as . Therefore,

@

the matrix converges to zero.

An equally important result is stated in the following theorem.

§CjG b¤

¥¦ ¥¡

Q £U

T

The series ¡

& '

&'()'

C B ¡B 3

converges if and only if . Under this condition, is nonsingular and the limit

0

3

of the series is equal to .

&

C

£

§ § ¥¦ T

The ¬rst part of the theorem is an immediate consequence of Theorem 1.4. In-

¤ '

7

deed, if the series converges, then . By the previous theorem, this implies that

Q C ©¡

B . To show that the converse is also true, use the equality

— )1 — 6 ™™— B C B 0 '

4 '

3 3

—

C

Q C B©¡ 3

and exploit the fact that since , then is nonsingular, and therefore,

' — )1 — 6 ™™— C 0 ' B 0 C 4

3 3

—

B &

0

3

B

This shows that the series converges since the left-hand side will converge to .

&

C

In addition, it also shows the second part of the theorem.

Another important consequence of the Jordan canonical form is a result that relates

the spectral radius of a matrix to its matrix norm.

y

nd¥ ¥ ¡ £ ¥ £ ”˜

uu d

u

¥ ¢ ¡ ¥ ¥ ¥ § ¥ ¤© ¡

¡

¥

§

©

¡ ¥

SG b¡¤

§¥¦ ¥

Q T

For any matrix norm , we have

¢ ' C B©¡

'F0 '

¡

¡£

¢

£

©¨¦

§ §¥ T

The proof is a direct application of the Jordan canonical form and is the subject

of Exercise 10.

£( ¢$© § § §

%§ £§ ) ©¨ £ ¤( § %

§

%

!

}{

| ¥

¤

This section examines speci¬c properties of normal matrices and Hermitian matrices, in-

cluding some optimality properties related to their spectra. The most common normal ma-

trices that arise in practice are Hermitian or skew-Hermitian.

¦¨

¦¥§¢

¡ ¦¡ ' 6 2' 6 )

§

1

§

By de¬nition, a matrix is said to be normal if it commutes with its transpose conjugate,

i.e., if it satis¬es the relation

t

I ( I ¨ ¤ & §¥

¦

An immediate property of normal matrices is stated in the following lemma.

„ § “£

¢ §¥ U

T

If a normal matrix is triangular, then it is a diagonal matrix.

£

©¨¦

§ §¥ T

Assume, for example, that is upper triangular and normal. Compare the ¬rst

diagonal element of the left-hand side matrix of (1.25) with the corresponding element of

the matrix on the right-hand side. We obtain that

6 … 0 ‚ 0 ‘ ( &… 6 E00 ‚

which shows that the elements of the ¬rst row are zeros except for the diagonal one. The

same argument can now be used for the second row, the third row, and so on to the last row,

H… ‚

a

7

to show that for .

A consequence of this lemma is the following important result.

SG b¡¤

§¥¦ ¥ Q ¡T A matrix is normal if and only if it is unitarily similar to a diagonal

matrix.

©¨¦£

§ §¥ T

It is straightforward to verify that a matrix which is unitarily similar to a diagonal