Open Access

Nonexpansive Matrices with Applications to Solutions of Linear Systems by Fixed Point Iterations

Fixed Point Theory and Applications20092010:821928

DOI: 10.1155/2010/821928

Received: 28 August 2009

Accepted: 19 October 2009

Published: 25 October 2009

Abstract

We characterize (i) matrices which are nonexpansive with respect to some matrix norms, and (ii) matrices whose average iterates approach zero or are bounded. Then we apply these results to iterative solutions of a system of linear equations.

Throughout this paper, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq1_HTML.gif will denote the set of real numbers, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq2_HTML.gif the set of complex numbers, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq3_HTML.gif the complex vector space of complex https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq4_HTML.gif matrices. A function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq5_HTML.gif is a matrix norm if for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq6_HTML.gif , it satisfies the following five axioms:

(1) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq7_HTML.gif ;

(2) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq8_HTML.gif if and only if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq9_HTML.gif ;

(3) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq10_HTML.gif for all complex scalars https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq11_HTML.gif ;

(4) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq12_HTML.gif ;

(5) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq13_HTML.gif .

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq14_HTML.gif be a norm on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq15_HTML.gif . Define https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq16_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq17_HTML.gif by

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_Equ1_HTML.gif
(1)

This norm on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq18_HTML.gif is a matrix norm, called the matrix norm induced by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq19_HTML.gif . A matrix norm on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq20_HTML.gif is called an induced matrix norm if it is induced by some norm on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq21_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq22_HTML.gif is a matrix norm on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq23_HTML.gif , there exists an induced matrix norm https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq24_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq25_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq26_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq27_HTML.gif (cf. [1, page 297]). Indeed one can take https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq28_HTML.gif to be the matrix norm induced by the norm https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq29_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq30_HTML.gif defined by

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_Equ2_HTML.gif
(2)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq31_HTML.gif is the matrix in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq32_HTML.gif whose columns are all equal to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq33_HTML.gif . For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq34_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq35_HTML.gif denotes the spectral radius of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq36_HTML.gif .

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq37_HTML.gif be a norm in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq38_HTML.gif . A matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq39_HTML.gif is a contraction relative to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq40_HTML.gif if it is a contraction as a transformation from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq41_HTML.gif into https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq42_HTML.gif ; that is, there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq43_HTML.gif such that

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_Equ3_HTML.gif
(3)

Evidently this means that for the matrix norm https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq44_HTML.gif induced by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq45_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq46_HTML.gif . The following theorem is well known (cf. [1, Sections 5.6.9–5.6.12]).

Theorem 1.

For a matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq47_HTML.gif , the following are equivalent:

(a) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq48_HTML.gif is a contraction relative to a norm in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq49_HTML.gif ;

(b) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq50_HTML.gif for some induced matrix norm https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq51_HTML.gif ;

(c) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq52_HTML.gif for some matrix norm https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq53_HTML.gif ;

(d) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq54_HTML.gif ;

(e) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq55_HTML.gif

That (b) follows from (c) is a consequence of the previous remark about an induced matrix norm being less than a matrix norm. Since all norms on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq56_HTML.gif are equivalent, the limit in (d) can be relative to any norm on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq57_HTML.gif , so that (d) is equivalent to all the entries of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq58_HTML.gif converge to zero as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq59_HTML.gif , which in turn is equivalent to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq60_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq61_HTML.gif .

In this paper, we first characterize matrices in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq62_HTML.gif that are nonexpansive relative to some norm https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq63_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq64_HTML.gif , that is,

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_Equ4_HTML.gif
(4)

Then we characterize those https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq65_HTML.gif such that

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_Equ5_HTML.gif
(5)

converges to zero as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq66_HTML.gif , and those that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq67_HTML.gif is bounded.

Finally we apply our theory to approximation of solution of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq68_HTML.gif using iterative methods (fixed point iteration methods).

Theorem 2.

For a matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq69_HTML.gif , the following are equivalent:

(a) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq70_HTML.gif is nonexpansive relative to some norm on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq71_HTML.gif ;

(b) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq72_HTML.gif for some induced matrix norm https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq73_HTML.gif ;

(c) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq74_HTML.gif for some matrix norm https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq75_HTML.gif ;

(d) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq76_HTML.gif is bounded;

(e) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq77_HTML.gif , and for any eigenvalue https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq78_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq79_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq80_HTML.gif , the geometric multiplicity is equal to the algebraic multiplicity.

Proof.

As in the previous theorem, (a), (b), and (c) are equivalent. Assume that (b) holds. Let the norm https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq81_HTML.gif be induced by a vector norm https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq82_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq83_HTML.gif . Then

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_Equ6_HTML.gif
(6)

proving that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq84_HTML.gif is bounded in norm https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq85_HTML.gif for every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq86_HTML.gif . Taking https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq87_HTML.gif , we see that the set of all columns of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq88_HTML.gif is bounded. This proves that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq89_HTML.gif is bounded in maximum column sum matrix norm ([1, page 294]), and hence in any norm in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq90_HTML.gif . Note that the last part of the proof also follows from the Uniform Boundedness Principle (see, e.g., [2, Corollary https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq91_HTML.gif , page 66])

Now we prove that (d) implies (e). Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq92_HTML.gif has an eigenvalue https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq93_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq94_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq95_HTML.gif be an eigenvector corresponding to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq96_HTML.gif . Then

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_Equ7_HTML.gif
(7)

as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq97_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq98_HTML.gif is any vector norm of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq99_HTML.gif . This contradicts (d). Hence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq100_HTML.gif . Now suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq101_HTML.gif is an eigenvalue with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq102_HTML.gif and the Jordan block corresponding to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq103_HTML.gif is not diagonal. Then there exist nonzero vectors https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq104_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq105_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq106_HTML.gif . Then

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_Equ8_HTML.gif
(8)

and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq107_HTML.gif . It follows that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq108_HTML.gif is unbounded, contradicting (d). Hence (d) implies (e).

Lastly we prove that (e) implies (c). Assume that (e) holds. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq109_HTML.gif is similar to its Jordan canonical form https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq110_HTML.gif whose nonzero off-diagonal entries can be made arbitrarily small by similarity ([1, page 128]). Since the Jordan block for each eigenvalue with modulus 1 is diagonal, we see that there is an invertible matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq111_HTML.gif such that the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq112_HTML.gif -sum of each row of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq113_HTML.gif is less than or equal to 1, that is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq114_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq115_HTML.gif is the maximum row sum matrix norm ([1, page 295]). Define a matrix norm https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq116_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq117_HTML.gif . Then we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq118_HTML.gif .

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq119_HTML.gif be an eigenvalue of a matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq120_HTML.gif . The index of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq121_HTML.gif , denoted by index( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq122_HTML.gif ) is the smallest value of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq123_HTML.gif for which https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq124_HTML.gif ([1, pages 148 and 131]). Thus condition (e) above can be restated as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq125_HTML.gif , and for any eigenvalue https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq126_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq127_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq128_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq129_HTML.gif .

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq130_HTML.gif . Consider

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_Equ9_HTML.gif
(9)

We call https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq131_HTML.gif the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq132_HTML.gif -average of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq133_HTML.gif . As with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq134_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq135_HTML.gif for every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq136_HTML.gif if and only if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq137_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq138_HTML.gif , and that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq139_HTML.gif is bounded for every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq140_HTML.gif if and only if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq141_HTML.gif is bounded in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq142_HTML.gif . We have the following theorem.

Theorem 3.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq143_HTML.gif . Then

(a) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq144_HTML.gif converges to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq145_HTML.gif if and only if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq146_HTML.gif for some matrix norm https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq147_HTML.gif and that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq148_HTML.gif is not an eigenvalue of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq149_HTML.gif ,

(b) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq150_HTML.gif is bounded if and only if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq151_HTML.gif for every eigenvalue https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq152_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq153_HTML.gif and that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq154_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq155_HTML.gif is an eigenvalue of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq156_HTML.gif .

Proof.

First we prove the sufficiency part of (a). Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq157_HTML.gif be a vector in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq158_HTML.gif . Let

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_Equ10_HTML.gif
(10)

By Theorem 2 for any eigenvalues https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq159_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq160_HTML.gif either https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq161_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq162_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq163_HTML.gif .

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq164_HTML.gif is written in its Jordan canonical form https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq165_HTML.gif , then the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq166_HTML.gif -average of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq167_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq168_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq169_HTML.gif is the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq170_HTML.gif -average of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq171_HTML.gif . https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq172_HTML.gif is in turn composed of the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq173_HTML.gif -average of each of its Jordan blocks. For a Jordan block of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq174_HTML.gif of the form

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_Equ11_HTML.gif
(11)

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq175_HTML.gif must be less than 1. Its https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq176_HTML.gif -average has constant diagonal and upper diagonals. Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq177_HTML.gif be the constat value of its https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq178_HTML.gif th upper diagonal ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq179_HTML.gif being the diagonal) and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq180_HTML.gif . Then ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq181_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq182_HTML.gif )

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_Equ12_HTML.gif
(12)

Using the relation https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq183_HTML.gif , we obtain

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_Equ13_HTML.gif
(13)

Thus, we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq184_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq185_HTML.gif . By induction, using (13) above and the fact that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq186_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq187_HTML.gif , we obtain https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq188_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq189_HTML.gif . Therefore https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq190_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq191_HTML.gif .

If the Jordan block is diagonal of constant value https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq192_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq193_HTML.gif and the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq194_HTML.gif -average of the block is diagonal of constant value https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq195_HTML.gif .

We conclude that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq196_HTML.gif and hence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq197_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq198_HTML.gif .

Now we prove the necessity part of (a). If 1 is an eigenvalue of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq199_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq200_HTML.gif is a corresponding eigenvector, then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq201_HTML.gif for every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq202_HTML.gif and of course https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq203_HTML.gif fails to converge to 0. If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq204_HTML.gif is an eigenvalue of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq205_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq206_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq207_HTML.gif is a corresponding eigenvector, then

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_Equ14_HTML.gif
(14)

which approaches to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq208_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq209_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq210_HTML.gif is an eigenvalue of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq211_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq212_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq213_HTML.gif , then there exist nonzero vectors https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq214_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq215_HTML.gif . Then by using the identity

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_Equ15_HTML.gif
(15)

we get

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_Equ16_HTML.gif
(16)

It follows that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq216_HTML.gif does not exist. This completes the proof of part (a).

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq217_HTML.gif satisfies the conditions in (b) and that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq218_HTML.gif is the Jordan canonical form of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq219_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq220_HTML.gif be an eigenvalue of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq221_HTML.gif and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq222_HTML.gif be a column vector of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq223_HTML.gif corresponding to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq224_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq225_HTML.gif , then the restriction https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq226_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq227_HTML.gif to the subspace spanned by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq228_HTML.gif is a contraction, and we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq229_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq230_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq231_HTML.gif , then by conditions in (b) either https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq232_HTML.gif , or there exist https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq233_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq234_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq235_HTML.gif . In the former case, we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq236_HTML.gif and in the latter case, we see from (16) that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq237_HTML.gif is bounded. Finally if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq238_HTML.gif then since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq239_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq240_HTML.gif and hence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq241_HTML.gif . In all cases, we proved that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq242_HTML.gif is bounded. Since column vectors of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq243_HTML.gif form a basis for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq244_HTML.gif , the sufficiency part of (b) follows.

Now we prove the necessity part of (b). If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq245_HTML.gif has an eigenvalue https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq246_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq247_HTML.gif and eigenvector https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq248_HTML.gif , then as shown above https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq249_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq250_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq251_HTML.gif has 1 as an eigenvalue and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq252_HTML.gif , then there exist nonzero vectors https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq253_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq254_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq255_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq256_HTML.gif which is unbounded. If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq257_HTML.gif is an eigenvalue of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq258_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq259_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq260_HTML.gif , then there exist nonzero vectors https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq261_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq262_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq263_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq264_HTML.gif . By expanding https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq265_HTML.gif and using the identity

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_Equ17_HTML.gif
(17)

we obtain

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_Equ18_HTML.gif
(18)

which approaches to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq266_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq267_HTML.gif . This completes the proof.

We now consider applications of preceding theorems to approximation of solution of a linear system https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq268_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq269_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq270_HTML.gif a given vector in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq271_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq272_HTML.gif be a given invertible matrix in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq273_HTML.gif . https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq274_HTML.gif is a solution of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq275_HTML.gif if and only if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq276_HTML.gif is a fixed point of the mapping https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq277_HTML.gif defined by

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_Equ19_HTML.gif
(19)

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq278_HTML.gif is a contraction if and only if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq279_HTML.gif is. In this case, by the well known Contraction Mapping Theorem, given any initial vector https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq280_HTML.gif , the sequence of iterates https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq281_HTML.gif converges to the unique solution of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq282_HTML.gif . In practice, given https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq283_HTML.gif , each successive https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq284_HTML.gif is obtained from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq285_HTML.gif by solving the equation

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_Equ20_HTML.gif
(20)

The classical methods of Richardson, Jacobi, and Gauss-Seidel (see, e.g., [3]) have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq286_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq287_HTML.gif respectively, where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq288_HTML.gif is the identity matrix, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq289_HTML.gif the diagonal matrix containing the diagonal of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq290_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq291_HTML.gif the lower triangular matrix containing the lower triangular portion of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq292_HTML.gif . Thus by Theorem 1 we have the following known theorem.

Theorem 4.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq293_HTML.gif , with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq294_HTML.gif invertible. Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq295_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq296_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq297_HTML.gif is invertible and the sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq298_HTML.gif defined recursively by

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_Equ21_HTML.gif
(21)

converges to the unique solution of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq299_HTML.gif .

Theorem 4 fails if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq300_HTML.gif , For a simple https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq301_HTML.gif example, let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq302_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq303_HTML.gif any nonzero vector.

We need the following lemma in the proof of the next two theorems. For a matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq304_HTML.gif , we will denote https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq305_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq306_HTML.gif the range and the null space of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq307_HTML.gif respectively.

Lemma 5.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq308_HTML.gif be a singular matrix in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq309_HTML.gif such that the geometric multiplicity and the algebraic multiplicity of the eigenvalue 0 are equal, that is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq310_HTML.gif . Then there is a unique projection https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq311_HTML.gif whose range is the range of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq312_HTML.gif and whose null space is the null space of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq313_HTML.gif , or equivalently, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq314_HTML.gif . Moreover, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq315_HTML.gif restricted to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq316_HTML.gif is an invertible transformation from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq317_HTML.gif onto https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq318_HTML.gif .

Proof.

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq319_HTML.gif is a Jordan canonical form of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq320_HTML.gif where the eigenvalues 0 appear at the end portion of the diagonal of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq321_HTML.gif , then the matrix
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_Equ22_HTML.gif
(22)

is the required projection. Obviously https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq322_HTML.gif maps https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq323_HTML.gif into https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq324_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq325_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq326_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq327_HTML.gif and so https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq328_HTML.gif . This proves that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq329_HTML.gif is invertible on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq330_HTML.gif .

Remark 6.

Under the assumptions of Lemma 5, we will call the component of a vector https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq331_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq332_HTML.gif the projection of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq333_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq334_HTML.gif along https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq335_HTML.gif . Note that by definition of index, the condition in the lemma is equivalent to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq336_HTML.gif .

Theorem 7.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq337_HTML.gif be a matrix in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq338_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq339_HTML.gif a vector in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq340_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq341_HTML.gif be an invertible matrix in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq342_HTML.gif and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq343_HTML.gif . Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq344_HTML.gif and that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq345_HTML.gif for every eigenvalue https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq346_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq347_HTML.gif with modulus https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq348_HTML.gif , that is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq349_HTML.gif is nonexpansive relative to a matrix norm. Starting with an initial vector https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq350_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq351_HTML.gif define https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq352_HTML.gif recursively by

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_Equ23_HTML.gif
(23)

for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq353_HTML.gif Let

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_Equ24_HTML.gif
(24)

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq354_HTML.gif is consistent, that is, has a solution, then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq355_HTML.gif converge to a solution vector https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq356_HTML.gif with rate of convergence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq357_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq358_HTML.gif is inconsistent, then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq359_HTML.gif . More precisely, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq360_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq361_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq362_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq363_HTML.gif is the projection of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq364_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq365_HTML.gif along https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq366_HTML.gif .

Proof.

First we assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq367_HTML.gif is invertible so that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq368_HTML.gif is also invertible. Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq369_HTML.gif be the mapping defined by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq370_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq371_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq372_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq373_HTML.gif and hence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq374_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq375_HTML.gif . https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq376_HTML.gif is the unique solution of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq377_HTML.gif and

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_Equ25_HTML.gif
(25)

Since the sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq378_HTML.gif in the theorem is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq379_HTML.gif , we have

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_Equ26_HTML.gif
(26)

Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq380_HTML.gif is invertible, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq381_HTML.gif is not an eigenvalue of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq382_HTML.gif , and by Theorem 3 part (a) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq383_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq384_HTML.gif . Moreover, from the proof of the same theorem, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq385_HTML.gif .

Next we consider the case when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq386_HTML.gif is not invertible. Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq387_HTML.gif is invertible, we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq388_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq389_HTML.gif . The index of the eigenvalue 0 of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq390_HTML.gif is the index of eigenvalue 1 of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq391_HTML.gif . Thus by Lemma 5, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq392_HTML.gif . For every vector https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq393_HTML.gif , let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq394_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq395_HTML.gif denote the component of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq396_HTML.gif in the subspace https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq397_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq398_HTML.gif , respectively.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq399_HTML.gif is consistent, that is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq400_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq401_HTML.gif . By Lemma 5, the restriction of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq402_HTML.gif on its range is invertible, so there exists a unique https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq403_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq404_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq405_HTML.gif , or equivalently, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq406_HTML.gif . For any vector https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq407_HTML.gif , we have

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_Equ27_HTML.gif
(27)

Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq408_HTML.gif maps https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq409_HTML.gif into https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq410_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq411_HTML.gif restricted to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq412_HTML.gif is invertible, we can apply the preceding proof and conclude that the sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq413_HTML.gif as defined before converges to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq414_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq415_HTML.gif . Now https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq416_HTML.gif , showing that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq417_HTML.gif is a solution of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq418_HTML.gif .

Assume now that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq419_HTML.gif , that is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq420_HTML.gif is inconsistent. Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq421_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq422_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq423_HTML.gif As in the preceding case there exists a unique https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq424_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq425_HTML.gif . Note that for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq426_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq427_HTML.gif . Thus for any vector https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq428_HTML.gif and any positive integer https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq429_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_Equ28_HTML.gif
(28)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq430_HTML.gif . As in the preceding case, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq431_HTML.gif is bounded and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq432_HTML.gif converges to 0. Thus https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq433_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq434_HTML.gif , and hence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq435_HTML.gif . This completes the proof.

Next we consider another kind of iteration in which the nonlinear case was considered in Ishikawa [4]. Note that the type of mappings in this case is slightly weaker than nonexpansivity (see condition (c) in the next lemma).

Lemma 8.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq436_HTML.gif be an https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq437_HTML.gif matrix. The following are equivalent:

(a)for every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq438_HTML.gif , there exists a matrix norm https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq439_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq440_HTML.gif ,

(b)for every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq441_HTML.gif , there exists an induced matrix norm https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq442_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq443_HTML.gif ,

(c) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq444_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq445_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq446_HTML.gif is an eigenvalue of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq447_HTML.gif .

Proof.

As in the proof of Theorem 2, (a) and (b) are equivalent. For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq448_HTML.gif , denote https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq449_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq450_HTML.gif . Suppose now that (a) holds. Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq451_HTML.gif be an eigenvalue of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq452_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq453_HTML.gif is an eigenvalue of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq454_HTML.gif . By Theorem 2   https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq455_HTML.gif for every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq456_HTML.gif and hence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq457_HTML.gif . If 1 is an eigenvalue of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq458_HTML.gif , then it is also an eigenvalue of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq459_HTML.gif . By Theorem 2, the index of 1, as an eigenvalue of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq460_HTML.gif , is 1. Since obviously https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq461_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq462_HTML.gif have the same eigenvectors corresponding to the eigenvalue 1, the index of 1, as an eigenvalue of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq463_HTML.gif , is also 1. This proves (c).

Now assume (c) holds. Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq464_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq465_HTML.gif , every eigenvalue of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq466_HTML.gif , except possibly for 1, has modulus less than 1. Reasoning as above, if 1 is an eigenvalue of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq467_HTML.gif , then its index is 1. Therefore by Theorem 2, (a) holds. This completes the proof.

Theorem 9.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq468_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq469_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq470_HTML.gif be an invertible matrix in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq471_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq472_HTML.gif . Suppose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq473_HTML.gif and that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq474_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq475_HTML.gif is an eigenvalue of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq476_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq477_HTML.gif be fixed. Starting with an initial vector https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq478_HTML.gif , define https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq479_HTML.gif recursively by

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_Equ29_HTML.gif
(29)

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq480_HTML.gif is consistent, then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq481_HTML.gif converges to a solution vector https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq482_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq483_HTML.gif with rate of convergence given by

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_Equ30_HTML.gif
(30)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq484_HTML.gif is any number satisfying

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_Equ31_HTML.gif
(31)

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq485_HTML.gif is inconsistent, then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq486_HTML.gif ; more precisely,

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_Equ32_HTML.gif
(32)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq487_HTML.gif is the projection of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq488_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq489_HTML.gif along https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq490_HTML.gif .

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq491_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq492_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq493_HTML.gif .

First we assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq494_HTML.gif is invertible. Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq495_HTML.gif is invertible and 1 is not an eigenvalue of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq496_HTML.gif ; thus https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq497_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq498_HTML.gif . We have

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_Equ33_HTML.gif
(33)

By a well known theorem (see, e.g. [1]), https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq499_HTML.gif for every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq500_HTML.gif .

Assume now that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq501_HTML.gif is not invertible and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq502_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq503_HTML.gif is in the range of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq504_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq505_HTML.gif satisfies the condition in Lemma 8, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq506_HTML.gif satisfies the condition in Lemma 5. Thus the restriction of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq507_HTML.gif on its range is invertible and there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq508_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq509_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq510_HTML.gif , or equivalently, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq511_HTML.gif . For any vector https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq512_HTML.gif , we have

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_Equ34_HTML.gif
(34)

Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq513_HTML.gif maps https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq514_HTML.gif into https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq515_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq516_HTML.gif restricted to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq517_HTML.gif is invertible, we can apply the preceding proof and conclude that the sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq518_HTML.gif converges to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq519_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq520_HTML.gif . https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq521_HTML.gif solves https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq522_HTML.gif since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq523_HTML.gif .

Assume lastly that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq524_HTML.gif , that is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq525_HTML.gif is inconsistent. Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq526_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq527_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq528_HTML.gif . As before there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq529_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq530_HTML.gif . Note that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq531_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq532_HTML.gif . Then

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_Equ35_HTML.gif
(35)

Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq533_HTML.gif converges to 0, we have

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_Equ36_HTML.gif
(36)

and hence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq534_HTML.gif . This completes the proof.

By taking https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq535_HTML.gif and considering only nonexpansive matrices in Theorems 7 and 9, we obtain the following corollary.

Corollary 10.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq536_HTML.gif be an https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq537_HTML.gif matrix such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq538_HTML.gif for some matrix norm https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq539_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq540_HTML.gif be a vector in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq541_HTML.gif . Then:

(a) starting with an initial vector https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq542_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq543_HTML.gif define https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq544_HTML.gif recursively as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_Equ37_HTML.gif
(37)
for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq545_HTML.gif Let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_Equ38_HTML.gif
(38)
for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq546_HTML.gif If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq547_HTML.gif is consistent, then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq548_HTML.gif converges to a solution vector https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq549_HTML.gif with rate of convergence given by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_Equ39_HTML.gif
(39)

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq550_HTML.gif is inconsistent, then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq551_HTML.gif .

(b) let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq552_HTML.gif be a fixed number. Starting with an initial vector https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq553_HTML.gif , let

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_Equ40_HTML.gif
(40)
If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq554_HTML.gif is consistent, then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq555_HTML.gif converges to a solution vector https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq556_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq557_HTML.gif with rate of convergence given by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_Equ41_HTML.gif
(41)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq558_HTML.gif is any number satisfying
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_Equ42_HTML.gif
(42)

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq559_HTML.gif is inconsistent, then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq560_HTML.gif .

Remark 11.

If in the previous corollary, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq561_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq562_HTML.gif in part (b), the sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq563_HTML.gif converges to a solution. This is the Richardson method, see for example, [3]. Even in this case, our method in part (b) may yield a better approximation. For example if

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_Equ43_HTML.gif
(43)

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq564_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq565_HTML.gif , then the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq566_HTML.gif th iterate in the Richardson method is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq567_HTML.gif away from the solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq568_HTML.gif , while the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq569_HTML.gif th iterate using the method in the corollary part (b) with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq570_HTML.gif is less than https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq571_HTML.gif .

An https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq572_HTML.gif matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq573_HTML.gif is called diagonally dominant if

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_Equ44_HTML.gif
(44)

for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq574_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq575_HTML.gif is diagonally dominant with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq576_HTML.gif for every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq577_HTML.gif and if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq578_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq579_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq580_HTML.gif is the diagonal matrix containing the diagonal of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq581_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq582_HTML.gif the lower triangular matrix containing the lower triangular entries of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq583_HTML.gif , then it is easy to prove that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq584_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq585_HTML.gif denotes the maximum row sum matrix norm; see, for example, [1, 3]. The following follows from Theorems 7 and 9.

Corollary 12.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq586_HTML.gif be a diagonally dominant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq587_HTML.gif matrix with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq588_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq589_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq590_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq591_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq592_HTML.gif is the diagonal matrix containing the diagonal of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq593_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq594_HTML.gif the lower triangular matrix containing the lower triangular entries of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq595_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq596_HTML.gif be a vector in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq597_HTML.gif . Then:

(a) starting with an initial vector https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq598_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq599_HTML.gif define https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq600_HTML.gif recursively as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_Equ45_HTML.gif
(45)
for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq601_HTML.gif Let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_Equ46_HTML.gif
(46)
for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq602_HTML.gif If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq603_HTML.gif is consistent, then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq604_HTML.gif converges to a solution vector https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq605_HTML.gif with rate of convergence given by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_Equ47_HTML.gif
(47)

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq606_HTML.gif is inconsistent, then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq607_HTML.gif .

(b) Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq608_HTML.gif be a fixed number. Starting with an initial vector https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq609_HTML.gif , let

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_Equ48_HTML.gif
(48)
If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq610_HTML.gif is consistent, then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq611_HTML.gif converges to a solution vector https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq612_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq613_HTML.gif with rate of convergence given by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_Equ49_HTML.gif
(49)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq614_HTML.gif is any number satisfying
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_Equ50_HTML.gif
(50)

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq615_HTML.gif is inconsistent, then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F821928/MediaObjects/13663_2009_Article_1350_IEq616_HTML.gif .

Authors’ Affiliations

(1)
Department of Mathematical Sciences, George Mason University

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Copyright

© Teck-Cheong Lim. 2010

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.