Not every matrix that can be symmetrically permuted into a -matrix is consistently

¨§

¡ ©R

ordered. The important property here is that the partition preserves the order of the

indices of nonzero elements. In terms of the adjacency graph, there is a partition of

Y( E

the graph with the property that an oriented edge from to always points to a set V( E E

with a larger index if , or a smaller index otherwise. In particular, a very important

E

consequence is that edges corresponding to the lower triangular part will remain so in

the permuted matrix. The same is true for the upper triangular part. Indeed, if a nonzero

¬ eXX P uT G i © T P R G i c ¬ X u T R c X T

© ¥ T T

element in the permuted matrix is with , then by Y E ¤

¤ ¤

eX

X

E W “ ¡ E ¡¬

W “ ¡ ¡¢¬ ¡ E¡

de¬nition of the permutation , or . Because ¤

¤

of the order preservation, it is necessary that . A similar observation holds for the E

upper triangular part. Therefore, this results in the following proposition.

h¤ ¡ v A£

£¦ £¤B9

A

' ' ©' )$ )

§

§

If a matrix is consistently ordered, then there exists a permuta-

§§ ¥ T

¥ ¥

tion matrix such that is a -matrix and T

§ ¥¬ X ¥§ ¥ ()¥ § ¬ X §§ ¥ r±— ¯ i ¯®

¥

¥

¥

¤ ¤

in which represents the (strict) lower part of and the (strict) upper part of .

¤ ¤

T

With the above property it can be shown that for consistently ordered matrices the

X

eigenvalues of as de¬ned in Proposition 4.1 are also invariant with respect to .

¢

h¤ ¡ v A£

£¦ ¥¤B9

A

' ' ©' )$ )

§

Let be the Jacobi iteration matrix associated with a consistently

¢

§

ordered matrix , and let and be the lower and upper triangular parts of , respec-

£ ¤ ¢

´ ¡C

| ¢ Cv¥ ¡At ¢

| §| C

¡¨

tively. Then the eigenvalues of the matrix T

¬X y w

¢ £ ¤

¥

¬

de¬ned for do not depend on .

$

Y

T

£

X

6 A

£

¡

First transform into a -matrix using the permutation in (4.44) provided

¢

by the previous proposition T

¥ ¬ ¥X

¥ ¥ ¥ ¥

w

¢ £ ¤ )

y

¥ £ ¥ v¤

¬

¥ ¥

From the previous proposition, the lower part of is precisely . Simi- ¢ £

¬ ¤

¥ ¥

larly, the upper part is , the lower and upper parts of the associated -matrix.

¤ ¤

T T

Therefore, we only need to show that the property is true for a -matrix. X X

¬

In this case, for any , the matrix is similar to . This means that ¢ ¢ ¢

with being equal to

W “ @¤

¤¢ ¤

y

„

£

(

¬

¤

..

‚ .

W “X

¡ ¡

where the partitioning is associated with the subsets respectively.

(

(

W X

Note that -matrices and matrices with the structure (4.42) are two particular cases

of matrices which ful¬ll the assumptions of the above proposition. There are a number of

well known properties related to Property A and consistent orderings. For example, it is

possible to show that,

Property A is invariant under symmetric permutations.

¥

A matrix has Property A if and only if there is a permutation matrix such that

§ W “ ¢™¤ §

¥ ¥¬ is consistently ordered.

Consistently ordered matrices satisfy an important property which relates the eigenval-

ues of the corresponding SOR iteration matrices to those of the Jacobi iteration matrices.

The main theorem regarding the theory for SOR is a consequence of the following result

proved by Young [232]. Remember that T T

T

w ' # W “X % #

XT X #

# T ¬

² ²

£¡

¢ #

0 y

w 'W “ # # W “X % W “ # #

X

#