¬ ²

matrix since is nonnegative and . Therefore, is

¦ ¦ ¦

W “ W“

X y

² ¬W“ §

nonnegative as is . W“ ¦

W“

§

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

§ ²

nonnegative. Since and are nonsingular, the relation (4.35) shows again that is ¦

nonsingular and in addition, T

X

² T W “ §

¬

W “ W “

X

² T ¬

W “ W “ W “

r± a i ¯®

X

¦W “ ¦ ² ¬

T

¬

Clearly, is nonnegative by the assumptions, and as a result of the Perron-

¦ W “ X

©

Frobenius theorem, there is a nonnegative eigenvector associated with which is an ¦

eigenvalue, such that T

X

© ¦ ¬ ¦ ©

From this and by virtue of (4.36), it follows that T

XT

¦

§ ² y ¢©

¬ ©

X

W “

¦

© §

Since and are nonnegative, this shows that T

W “

XT

¦

YX

²

T T

¦

y

X ¥¦X

T

² ¬

and this can be true only when . Since is nonsingular, then ,

Y ¦ ¦ ¦

X

y y

which implies that . ¦

y

This theorem establishes that the iteration (4.34) always converges, if is a regu-

(

§

lar splitting and is an M-matrix.

¨6§ & 8 ©' 8 B©

E ©& ¡2DB 078 I 6C8 E B ©' ©

I

AE

F5) 3 A

§

We begin with a few standard de¬nitions.

¥ h¤

§ ' )$ 2¨) 2 ¤B9

£A

)§

¦

§

A matrix is

(weakly) diagonally dominant if

«g q £Rf

0 u u c0 (0 uR c0 }

¬ r¤

‘ (

(

y

Is

ih

hs

³ ¡C

| ¢ Cv¥ ¡At ¢

| §|

¡¨

strictly diagonally dominant if

£R

«

f

( 0 vR c 0 g q 0 u u c 0 ¬} W¤

u ‘ (V(

y

ipds

h

hs

§ £R

irreducibly diagonally dominant if is irreducible, and

«

f

( 0 vR c 0 g q 0 u u c 0 ¬} W¤

u ‘ (V(

y

ipds

h

hs

with strict inequality for at least one .

Often the term diagonally dominant is used instead of weakly diagonally dominant.

Diagonal dominance is related to an important result in Numerical Linear Algebra

known as Gershgorin™s theorem. This theorem allows rough locations for all the eigenval-

§

ues of to be determined. In some situations, it is desirable to determine these locations

§

in the complex plane by directly exploiting some knowledge of the entries of the matrix .

The simplest such result is the bound

§

0 R 0©