- About this Journal ·
- Abstracting and Indexing ·
- Advance Access ·
- Aims and Scope ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

Advances in Numerical Analysis

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

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

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

^{1}Department of Mathematics and Statistics, Qinghai University for Nationalities, Xining 810007, China^{2}School 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).

#### References

- J. M. Varah, “A lower bound for the smallest singular value of a matrix,”
*Linear Algebra and Its Applications*, vol. 11, pp. 3–5, 1975. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. S. Varga, “On diagonal dominance arguments for bounding ${\parallel {A}^{-1}\parallel}_{\infty}$,”
*Linear Algebra and Its Applications*, vol. 14, no. 3, pp. 211–217, 1976. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G.-H. Cheng and T.-Z. Huang, “An upper bound for ${\parallel {A}^{-1}\parallel}_{\infty}$ of strictly diagonally dominant $M$-matrices,”
*Linear Algebra and Its Applications*, vol. 426, no. 2-3, pp. 667–673, 2007. View at Publisher · View at Google Scholar · View at MathSciNet - P. Wang, “An upper bound for ${\parallel {A}^{-1}\parallel}_{\infty}$ of strictly diagonally dominant $M$-matrices,”
*Linear Algebra and Its Applications*, vol. 431, no. 5–7, pp. 511–517, 2009. View at Publisher · View at Google Scholar · View at MathSciNet - R. A. Horn and C. R. Johnson,
*Topics in Matrix Analysis*, Cambridge University Press, Cambridge, Mass, USA, 1991. View at Publisher · View at Google Scholar · View at MathSciNet - C. R. Johnson, “A Hadamard product involving $M$-matrices,”
*Linear and Multilinear Algebra*, vol. 4, no. 4, pp. 261–264, 1977. View at Google Scholar · View at MathSciNet - M. Fiedler, C. R. Johnson, and T. L. Markham, “A trace inequality for
*M*-matrices and the symmertrizability of a real matrix by a positive diagonal matrix,”*Linear Algebra and Its Applications*, vol. 102, pp. 1–8, 1988. View at Google Scholar - P. N. Shivakumar, J. J. Williams, Q. Ye, and C. A. Marinov, “On two-sided bounds related to weakly diagonally dominant $M$-matrices with application to digital circuit dynamics,”
*SIAM Journal on Matrix Analysis and Applications*, vol. 17, no. 2, pp. 298–312, 1996. View at Publisher · View at Google Scholar · View at MathSciNet - A. Berman and R. J. Plemmons,
*Nonnegative Matrices in the Mathematical Sciences*, Academic Press, New York, NY, USA, 1994. - Y. L. Zhang, H. M. Mo, and J. Z. Liu, “$\alpha $-diagonal dominance and criteria for generalized strictly diagonally dominant matrices,”
*Numerical Mathematics*, vol. 31, no. 2, pp. 119–128, 2009. View at Google Scholar · View at MathSciNet - P. N. Shivakumar and K. H. Chew, “A sufficient condition for nonvanishing of determinants,”
*Proceedings of the American Mathematical Society*, vol. 43, pp. 63–66, 1974. View at Publisher · View at Google Scholar · View at MathSciNet - S. Xu,
*Theory and Methods about Matrix Computation*, Tshua University Press, Beijing, China, 1986.