Advances in Numerical Analysis

Volume 2013 (2013), Article ID 980615, 6 pages

http://dx.doi.org/10.1155/2013/980615

Research Article

A New Upper Bound for of a Strictly -Diagonally Dominant -Matrix

Zhanshan Yang,¹ Bing Zheng,² and Xilan Liu¹

¹Department of Mathematics and Statistics, Qinghai University for Nationalities, Xining 810007, China

²School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China

Received 22 September 2013; Accepted 10 December 2013

Academic Editor: Ting-Zhu Huang

Copyright © 2013 Zhanshan Yang et al. 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.

Abstract.

A new upper bound for of a real strictly diagonally dominant -matrix is present, and a new lower bound of the smallest eigenvalue of is given, which improved the results in the literature. Furthermore, an upper bound for of a real strictly -diagonally dominant -matrix is shown.

1. Introduction

The estimation for the bound for the norm of a real invertible matrix is important in numerical analysis, so many researchers were devoted to studying this kind of problems. For example, Varah [1] discussed the bound for the infinity norm of a strictly diagonally dominant matrix and obtained the following estimation:

After that Varga [2] extended the result of [1] to -matrices. Evidently, the upper bound for in (1) only involves the entries in the matrix . If the diagonal dominance of is weak, that is, is small, then the bound given by (1) may be large. For this reason, some authors were devoted to improving the result of (1). Recently, Cheng and Huang [3] presented a more compacted upper bound for a strictly diagonally dominant -matrix

and then Wang [4] further improved this bound and gave the following result:

where notations in (2) and (3) have the same meanings as those used in this paper, which will be shown later.

In this paper, we present a new upper bound of a strictly diagonally dominant matrix , which is better than that obtained by Wang, and a new lower bound of the smallest eigenvalue of is also obtained. In addition, an upper bound for of a strictly -diagonal dominant matrix is presented. To our knowledge, little has been done for upper bound of strictly -diagonal dominant matrices. Further, examples are given to illustrate the performance of our results.

Next, we introduce some notations and definitions. As usual, let be an identity matrix of order . If there exists an nonnegative matrix and a real number such that with , then is called a nonsingular -matrix, where is the spectral radius of the nonnegative matrix . It is well known that the inverse matrix of a -matrix is nonnegative and, therefore, is a positive eigenvalue of related to the Perron eigenvalue of the nonnegative matrix . If denotes the minimum of the real parts of the eigenvalues of , that is, , then . For further properties of the -matrix , we refer the readers to [5–7].

An matrix is called a strictly diagonally dominant matrix if for . Let

where is the set of positive integers. For an matrix , the principal matrix of formed by rows and columns with indices between and is denoted by .

Definition 1 (see [8]). is weakly chained diagonally dominant if, for all , and and for all , , there exist indices in with , , where and .

Definition 2 (see [9]). Let , is strictly diagonally dominant if .

Obviously, if is a strictly diagonally dominant matrix, then be a weakly chained diagonally dominant matrix.

Definition 3 (see [9]). is an -matrix if, for all with , and .

Definition 4 (see [10]). Let ; if there exist , such that

for all , then is said to be an -diagonal dominant matrix, denoted by .

Remark 5. By Definition 4, we know that is just a diagonal dominant matrix while .

Definition 6. If all the inequalities in (5) strictly hold, then is said to be strictly -diagonal dominant matrix ().

2. Estimation for an Upper Bound for of Strictly Diagonally Dominant -Matrix

We state some lemmas before giving a new upper bound for .

Lemma 7 (see [3]). Let be an weakly chained diagonally dominant -matrix, , , and . Then, for ,

where

Furthermore, if , then .

Lemma 8 (see [11]). A weakly chained diagonally dominant -matrix is a nonsingular -matrix.

Lemma 9 (see [11]). Let be an weakly chained diagonally dominant -matrix; then is an weakly chained diagonally dominant M-matrix; that is, exists and .

Lemma 10 (see [11]). Let be an weakly chained diagonally dominant -matrix, . Then, for ,

Lemma 11 (see [11]). Let be an row strictly diagonally dominant -matrix; then

Lemma 12 (see [2]). Let be an row strictly diagonally dominant -matrix; then, for , we have

Lemma 13 (see [1]). Let be an weakly chained diagonally dominant M-matrix, , and , . Then

where

Now we give an upper bound for and of a strictly diagonally dominant -matrix by the following theorem.

Theorem 14. Let be an row strictly diagonally dominant M-matrix, . Then

Proof. We prove this theorem by induction.

(1)Let , , , and . Then

By Lemmas 7, 11, and 12, we know that

Let . By (8) and the second equality in (6), we have

From (8) with , we have

Thus, for , we obtain

So by (15) and (18), we get

(2)Applying induction with respect to of in (19) finishes the proof.

From Theorem 14 and Lemma 13, the following theorem can be obtained easily.

Theorem 15. Let be an row strictly diagonally dominant -matrix. Then the smallest eigenvalue of is

Theorem 16. Let be an row strictly diagonally dominant -matrix. Then the bound in (13) is sharper than that in (3), that is,

Proof. Since is a strictly diagonally dominant matrix, , , and , then we have

The results follow Lemma 12. Inequality (21) shows that the bound in (13) is better than that in (3).

For all , , we have

With the help of the above discussions, we give the upper bound for of a real strictly -diagonally dominant -matrix.

3. Estimation for an Upper Bound for of a Strictly -Diagonally Dominant -Matrix

We show some notations and lemmas which are necessary to our conclusions.

Lemma 17 (see [12]). Let , , be nonsingular, then

Lemma 18. Let is a strictly diagonal dominant -matrix. If , with

and if

then , where

Proof. By Theorem 14, we get

It is easy to see that , if

where

Lemma 19 (see [12]). If  , then is nonsingular and

Theorem 20. Let be a strictly -diagonal dominant matrix, , and be an -matrix. If, for those , , and , then

where

Proof. Note that . Then

So we can split , such that , where and

We know and is an -matrix. Thus, is a strictly diagonal dominant -matrix; hence, . Let , . If , by Lemma 18, we get . By Lemmas 17 and 19 and Theorem 14, we can obtain

Let .

Then

Further, we have

where

The proof is complete.

4. Examples

We illustrate our results by the following two examples.

(1)Consider the bound for of a strictly diagonal dominant matrix , where

Direct calculation by MATLAB R2010a gives

It is obvious that the bound of Theorem 14 of this paper is better than other known ones. Furthermore, we can estimate by Theorem 15.

(2)Consider the bound for of a strictly -diagonal dominant matrix for ,

Note that

We know that is not a strictly diagonal dominant matrix, and the bound of cannot be obtained by (2) or (3), but it can be estimated by (32) in Theorem 20.

Split the matrix such that , where and , , Then

The bound for can be estimated by (13) in Theorem 14 and (32) in Theorem 20 as follows:

Conflict of Interests

There is no conflict of interests regarding the publication of this paper.

Acknowledgment

This paper is supported by the NNSF of China (11171371, 11361047) and the NSF of Qinghai Province (2012-Z-910).

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