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Advances in Numerical Analysis
Volume 2013 (2013), Article ID 980615, 6 pages
A New Upper Bound for of a Strictly -Diagonally Dominant -Matrix
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.
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.
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  discussed the bound for the infinity norm of a strictly diagonally dominant matrix and obtained the following estimation: After that Varga  extended the result of  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  presented a more compacted upper bound for a strictly diagonally dominant -matrix and then Wang  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 ). is weakly chained diagonally dominant if, for all , and and for all , , there exist indices in with , , where and .
Definition 2 (see ). 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 ). is an -matrix if, for all with , and .
Definition 4 (see ). 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 ). Let be an weakly chained diagonally dominant -matrix, , , and . Then, for , where Furthermore, if , then .
Lemma 8 (see ). A weakly chained diagonally dominant -matrix is a nonsingular -matrix.
Lemma 9 (see ). Let be an weakly chained diagonally dominant -matrix; then is an weakly chained diagonally dominant M-matrix; that is, exists and .
Lemma 10 (see ). Let be an weakly chained diagonally dominant -matrix, . Then, for ,
Lemma 11 (see ). Let be an row strictly diagonally dominant -matrix; then
Lemma 12 (see ). Let be an row strictly diagonally dominant -matrix; then, for , we have
Lemma 13 (see ). 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.
Theorem 15. Let be an row strictly diagonally dominant -matrix. Then the smallest eigenvalue of 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 ). 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 ). 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
Then Further, we have where The proof is complete.
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
Conflict of Interests
There is no conflict of interests regarding the publication of this paper.
This paper is supported by the NNSF of China (11171371, 11361047) and the NSF of Qinghai Province (2012-Z-910).
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