Inequalities for the Minimum Eigenvalue of Doubly Strictly Diagonally Dominant -Matrices

Abstract

Let be a doubly strictly diagonally dominant -matrix. Inequalities on upper and lower bounds for the entries of the inverse of are given. And some new inequalities on the lower bound for the minimal eigenvalue of and the corresponding eigenvector are presented to establish an upper bound for the -norm of the solution for the linear differential system , .

1. Introduction

For a positive integer , denotes the set . For , we write () if all (), . If (), we say is nonnegative (positive, resp.). Let denote the class of all real matrices all of whose off-diagonal entries are nonpositive. A matrix is called an -matrix [1] if and the inverse of , denoted by , is nonnegative.

Let be an -matrix. Then there exist a positive eigenvalue of , , and a corresponding eigenvector , where is the Perron eigenvalue of the nonnegative matrix , , and denotes the spectrum of . is called the minimum eigenvalue of [2, 3]. If, in addition, is irreducible, then and is simple and , which is unique if we assume that the -norm of equals ; that is, [3]. If is the diagonal matrix of an -matrix and , then the spectral radius of the Jacobi iterative matrix of is denoted by . For a set , we denote by the cardinality of . Note that if and only if .

For convenience, we employ the following notations throughout. Let be nonsingular with , for all , and . We denote, for any ,

Definition 1 (see [4]). A matrix is called(i)(strictly) diagonally dominant, if (, resp.) for all , and is called doubly (strictly) diagonally dominant if (, resp.) for all ;(ii)weakly chained diagonally dominant, if , and for all , there exist indices in with , , where and .

Remark. (i) It is well known that a doubly strictly diagonally dominant matrix is nonsingular and that [5]. If , we denote by the unique element throughout; that is, . Meanwhile, if is doubly strictly diagonally dominant and , then is strictly diagonally dominant.

(ii) It is clear that a strictly diagonally dominant matrix is doubly strictly diagonally dominant and also weakly chained diagonally dominant. Also clearly, for a doubly strictly diagonally dominant matrix , if , then is weakly chained diagonally dominant; otherwise, is not weakly chained diagonally dominant.

Estimating the bounds of the minimum eigenvalue of an -matrix and its corresponding eigenvector is an interesting subject in matrix theory and has important applications in many practical problems; see [4, 6–8]. In particular, these bounds are used to estimate upper bounds of the -norm of the solution for the following system of ordinary differential equations: where , , and is a constant -matrix. And it is proved in [6] that where and is the positive eigenvector of corresponding to . When the order of is large, it is difficult to compute and . Hence it is necessary to estimate the bounds of and .

In [4], Shivakumar et al. obtained the following bounds of when is a weakly chained diagonally dominant -matrix.

Theorem 2 (see [4, Theorem  4.1]). Let be a weakly chained diagonally dominant -matrix and . Then

Recently, Tian and Huang [9] provided lower bounds of by using the spectral radius of the Jacobi iterative matrix for a general -matrix .

Theorem 3 (see [9, Theorem  3.1]). Let be an -matrix and . Then

Also in [9], a lower bound of , which depends only on the entries of , has been presented when is a strictly diagonally dominant -matrix.

Theorem 4 (see [9, Corollary  3.4]). Let be a strictly diagonally dominant -matrix. Then

As shown in [9], it is possible that equals zero or that is very small, and moreover, whenever is not weakly chained diagonally dominant, Theorems 2 and 4 cannot be used to estimate the bounds of effectively. On the other hand, it is difficult to estimate by using Theorem 3 because of the difficulty of computing the diagonal elements of and when is very large.

In this paper, we continue to research the problems mentioned previously. For a doubly strictly diagonally dominant -matrix , we in Section 3 give some inequalities on the bounds of the entries of . And in Section 4, some inequalities on bounds of and the corresponding eigenvector are established. Lastly, an example, in which we estimate the -norm of the solution for the system (2) when is a doubly strictly diagonally dominant -matrix, is given in Section 5.

2. Preliminaries

In this section, we give a lemma which involves some results for a doubly strictly diagonally dominant -matrix. First, some notations are listed: for a doubly strictly diagonally dominant matrix and , where Note here that let if ().

Lemma 5. Let be a doubly strictly diagonally dominant -matrix and . And, for any , let , where and , . Then is a strictly diagonally dominant -matrix. Furthermore, , for and for any .

Proof. Since is a doubly strictly diagonally dominant -matrix and , we have hence, from , And, for any , if , and if , inequality (11) is obvious.
From inequality (11), we have Let . Then From inequality (10), we have And, for any , from inequality (12), we have From inequality (14) and inequality (15), is strictly diagonally dominant. Moreover, it is clear that and , which implies that is an -matrix.
Furthermore, from the definition of , we have that and for any , We now prove for any . Since is doubly strictly diagonally dominant, we get that there is , , such that (otherwise, a contradiction to the definition of doubly strictly diagonally dominant matrices). Hence and equivalently, And for any , Hence, from inequality (19), inequality (20), and the fact that is an -matrix, we have that, for any , The proof is completed.

Lemma 6 (see [10, Page 719]). Let be an complex matrix and let be positive real numbers. Then all the eigenvalues of lie in the

3. Bounds for the Entries of the Inverse of a Doubly Strictly Diagonally Dominant -Matrix

In this section, upper and lower bounds for the entries of are given when is a doubly strictly diagonally dominant -matrix.

Lemma 7 (see [11, Lemma  2.2]). Let be a strictly diagonally dominant -matrix and let . Then, for all ,

Next, we present a similar result for a doubly strictly diagonally dominant -matrix.

Theorem 8. Let be a doubly strictly diagonally dominant -matrix and let . Then, for all ,

Proof. If , then is strictly diagonally dominant and the conclusion follows from Lemma 7. We next suppose that . From Lemma 5, we get that is a strictly diagonally dominant -matrix for any , where , , and , . Let and . Then If , from Lemma 7, we have that that is, If and , from Lemma 7, then that is, moreover, by , we have And if and , from Lemma 7, then that is, Hence, from inequality (30) and inequality (32) and letting , we have that, for any , The conclusion follows from inequality (27) and inequality (33).

We next establish the upper and lower bounds for the diagonal entries of the inverse of a doubly strictly diagonally dominant -matrix.

Theorem 9. Let be a doubly strictly diagonally dominant -matrix and let . Then, for all ,

Proof. If , then the conclusion follows from Lemma  2.2 of [9]. We next suppose that . Since is a doubly strictly diagonally dominant -matrix, and , , . By , we have that, for all , which implies Moreover, from equality (35) and Theorem 8, we have that, for any , And similar to the proof of Theorem 8, is a strictly diagonally dominant -matrix, where is given in Lemma 5. Let . Then, from , we have that Hence, from inequality (37), inequality (38), and Lemma 5, we obtain that for any The conclusion follows from inequality (36) and inequality (39).

Next a lower bound of the entries of the inverse of a doubly strictly diagonally dominant -matrix will be established. Firstly, a lemma is given.

Lemma 10 (see [4, Theorem  3.5]). Let be a weakly chained diagonally dominant -matrix and let . Then

Theorem 11. Let be a doubly strictly diagonally dominant -matrix and let . Then where

Proof. If , then is a strictly diagonally dominant -matrix, also a weakly chained diagonally dominant -matrix. The conclusion is evident from Lemma 10. We next suppose that . Similar to the proof of Theorem 8, is a strictly diagonally dominant -matrix, where is given in Lemma 5. Let and . By Lemma 10, we have that Moreover, note that and ; we have And also note that, for any , Hence, we need only prove that for any . In fact, if , then If , then If , then Hence, for any , The conclusion follows from inequalities (44), (45), and (49).

4. Bounds for the Minimum Eigenvalue of a Doubly Strictly Diagonally Dominant -Matrix

In this section, we give some lower bounds for which depend only on the entries of when is a doubly strictly diagonally dominant -matrix. First, for , we give an upper bound of , where .

Theorem 12. Let be a doubly strictly diagonally dominant -matrix. Then

Proof. Let . Then The proof is completed.

Theorem 13. Let be a doubly strictly diagonally dominant -matrix. Then

Proof. If is irreducible, then ; meanwhile, from the irreducibility of and the definition of , we have for any . We next consider the spectral radius of . From Lemma 6, we have that there is such that which, from [12], leads to Hence,
If is reducible, then we can obtain a doubly strictly diagonally dominant -matrix such that is irreducible by replacing some nondiagonal zero entries of with sufficiently small negative real number . Now replace with in the previous case. Let approach 0; the conclusion follows by the continuity of about the entries of .

From Theorems 12 and 13, we have the following result.

Theorem 14. Let be a doubly strictly diagonally dominant -matrix. Then where

Proof. By Theorem 12 and the fact that , we have that Hence, from Theorem 13, .

We now give upper and lower bounds for the components of the eigenvector corresponding to the minimum eigenvalue for an irreducible doubly strictly diagonally dominant -matrix.

Theorem 15. Let be an irreducible doubly strictly diagonally dominant -matrix and let . And let be the positive eigenvector of corresponding to with . Then, for all , Furthermore,

Proof. It is clear that exists and . From and , we have and ; hence, where . The lower bound for is proved similarly. Furthermore, by Theorem  3.1 of [12], By Theorem 8, . Hence, The proof is completed.

5. Example

Consider the following system: where It is easy to verify that is an irreducible doubly strictly diagonally dominant -matrix and that . Hence is not a weakly chained diagonally dominant -matrix. We now establish the upper bound for the -norm of the solution . Let . By Theorems 8 and 9, we have By Theorem 11, we have By Theorem 14, we have Hence, by inequality (3) and Theorem 15, we have Hence, Note here that we cannot estimate the lower bound of by using Theorem 2 (Theorem  4.1 of [4]) and Theorem 4 (Corollary  3.4 of [9]) because is not a strictly diagonally dominant -matrix and not a weakly chained diagonally dominant -matrix.

Conflict of Interests

The authors declare that they have no conflict of interests.

Authors’ Contribution

Ming Xu, Suhua Li, and Chaoqian Li contributed equally to this work. All authors read and approved the final paper.

Acknowledgments

The authors are grateful to the referees for their useful and constructive suggestions. The first author is supported by Science Foundation of Guizhou Province (20132260, LKK201331, and LKK201424). The third author is supported by National Natural Science Foundations of China (11326242, 11361074), Natural Science Foundations of Yunnan Province (2013FD002), and IRTSTYN.