£

©¨¦

§ §¥ T 7

The proof is by induction. The inequality is clearly true for . Assume that

(1.35) is true for . According to the previous proposition, multiplying (1.35) from the left

by results in

0 4' t

X ' ¨ ¢ ¦§¥

˜

#

uY u d¥ £ ¡ ¨ ¢¤¥ ¤¤§ ¥ §¢

n ¦¢¥ £ ¡© ¡

¥ ¡ © ¥ ¡ ¥ © ¥ ¥

¥ ¨

('˜ ' ( ˜ 7 7

Now, it is clear that if , then also , by Proposition 1.6. We now multiply both

˜ # ˜

sides of the inequality by to the right, and obtain

4

t

0 ' ˜ # ' X ¨ ¢¢ §¥

˜ (¦

4 0 4'

0'

˜s#

The inequalities (1.36) and (1.37) show that , which completes the induction

proof.

A theorem which has important consequences on the analysis of iterative methods will

now be stated.

„ §CjG b¤

¥¦ ¥¡

Q U

T

˜

Let and be two square matrices that satisfy the inequalities

t

˜s# s# ¡ ¨ ¢¢ §¥

2¦

Then

C ˜ B©¡ # C B©¡ t

¨ ¢ ¢ §¥

¦

£

§ §¦

¥ T

The proof is based on the following equality stated in Theorem 1.6

B¡

'F0 ' ¢

¡

' C0

0 ¡¢

3

for any matrix norm. Choosing the norm, for example, we have from the last property

in Proposition 1.6

B ¡ 'F00 ' ˜ ¢ # ' 00 ' ¢ B ¡

¡ ¡

˜ ¡¢ ¡¢

'C

' C

which completes the proof.

„ §CjG b¤

¥¦ ¥¡ C ˜ B ¡

Q £U

T 3

˜˜ ˜

Let be a nonnegative matrix. Then if and only if

0

3

B

is nonsingular and is nonnegative.

&

C

§ § ¥¦£ C ˜B ¡

– T 3 ˜

De¬ne . If it is assumed that , then by Theorem 1.5,

˜ –

3 is nonsingular and ¡

˜ ( & 0 C ˜ t

0 – 3 ¨ £ £ §¥

B ¦

&

&

&

(˜ ˜

7

In addition, since , all the powers of as well as their sum in (1.40) are also

nonnegative.

–

To prove the suf¬cient condition, assume that is nonsingular and that its inverse

is nonnegative. By the Perron-Frobenius theorem, there is a nonnegative eigenvector V

˜ ©¡

B

associated with , which is an eigenvalue, i.e.,

C

˜¡ ˜ B

I

V V

C

or, equivalently,

˜ ©¡ 0 –

& V V

B 3

˜B ¡

C

0 3

– ˜

c

3 7

Since and are nonnegative, and is nonsingular, this shows that ,

V S

&

C

which is the desired result.

d “ ©¢ d¡ ¥ © ”¥ 7© ¥ ¥ ¨¥ £

e u § u§ ¨ £

u

¡ ¡£ ¢ ¡ ¥ 5 ¥

§¥

©

§ ¥

¡

¤q ¥

SRP I#GDE C

H QF F HF U

T

A matrix is said to be an -matrix if it satis¬es the following four

properties:

˜}ia)`1x1Y # „‚‚

"

Yy

7

for .

#

˜ i)11Y

"… 7