and

‘£ p p‘ pY¢ uX ‘‘ ‘ z “¢ ¤¥£ £¢ p¢ ‘ ‘z — ¤¥£ ¢ u 3

uX

8 ‘ p 7 7 3 ¤7 7¤ 7 3 3¦ 7

¢

u

A ¤£ qz 7 7

3 7 3 7 3 ¡

7

5

¢

7 7 ¡7

It is possible to avoid complex arithmetic by using the quasi-Schur form which consists of

the pair of matrices

¥ ¢ ¤ qz7 £ %qz7 ££ ¤ ¥ pW3

£ zW3 ¢ z7 z%p7 p7W3

7

3 8

£ z7 z ¥ 7 ¤£ z7 z p7

¢ uY 7

© 5 A

§ ¤

¤

and

‘3 ¤ ¨ 7 77 ¤ ¤ 3 ¤¥ %¥ ¢ Y

Y

¤ 7 ¢ 7 z£

3 8

7¤

5 A

¢

5

£ ¡qz7 7

7

We conclude this section by pointing out that the Schur and the quasi-Schur forms

of a given matrix are in no way unique. In addition to the dependence on the ordering

©

0 ¤

of the eigenvalues, any column of can be multiplied by a complex sign and a new

corresponding can be found. For the quasi-Schur form, there are in¬nitely many ways

5

to select the blocks, corresponding to applying arbitrary rotations to the columns of

0 associated with these blocks.

¦ ¦ IP '

¥¤¢

¡ £¡ ! ¢P 10'

' ¢#

6 !4 §¨

3 ¨§

The analysis of many numerical techniques is based on understanding the behavior of the

'

successive powers of a given matrix . In this regard, the following theorem plays a

fundamental role in numerical linear algebra, more particularly in the analysis of iterative

methods.

S¥ G b¡B©¤¡

§¦¥ 511)‘)z

'

Q T

7

The sequence , converges to zero if and only if

.

C

£ ¤'

©¨¦

§ §¥ T

7 0V

To prove the necessary condition, assume that and consider a unit

0@

eigenvector associated with an eigenvalue of maximum modulus. We have

0 ' 0

'

0@

V V

which implies, by taking the 2-norms of both sides,

' ' p

¤ 60

0 7

@ V

uY u d¥ £ ¡ ¨ ¢¤¥ ¤¤§ ¥ §¢

n ¦¢¥ £ ¡© ¡

¥ ¡ © ¥ ¡ ¥ © ¥ ¥

¥ ¨

Q 0 C ©¡

B

This shows that . @

The Jordan canonical form must be used to show the suf¬cient condition. Assume that

Q C ©¡

B . Start with the equality

0 ' 0

'

0 &

' '

To prove that converges to zero, it is suf¬cient to show that converges to zero. An

'

important observation is that preserves its block form. Therefore, it is suf¬cient to prove

that each of the Jordan blocks converges to zero. Each block is of the form

¢ — @Y

@

Q (

¢ ! ¢ Q 78

¡

where is a nilpotent matrix of index , i.e., . Therefore, for ,

0!

… ¢ … ' ¡

¢

& '

&

q B (…

&

@

¢ ¢

3

&'

C

Q (

Using the triangle inequality for any norm and taking yields

0!

… … '

£ ¢

&

¢ '

&

#