- Research
- Open access
- Published:
Some new bounds for the minimum eigenvalue of the Hadamard product of an M-matrix and an inverse M-matrix
Journal of Inequalities and Applications volume 2013, Article number: 480 (2013)
Abstract
Let A and B be nonsingular M-matrices. Several new bounds on the minimum eigenvalue for the Hadamard product of B and the inverse matrix of A are given. These bounds can improve considerably some previous results.
MSC:15A42, 15B34.
1 Introduction
Let () denote the set of all complex (real) matrices, , . We write if for any . If , A is called a nonnegative matrix. The spectral radius of A is denoted by .
We denote by the class of all real matrices, whose off-diagonal entries are nonpositive. A matrix is called a nonsingular M-matrix if there exist a nonnegative matrix B and a nonnegative real number s such that with , where I is the identity matrix. will be used to denote the set of all nonsingular M-matrices. Let us denote , where denotes the spectrum of A.
The Hadamard product of two matrices and is the matrix . If , then is also an M-matrix (see [1]).
Let be an matrix with all diagonal entries being nonzero throughout. For , , denote
In 2013, Zhou et al. [2] obtained the following result: If is a strictly row diagonally dominant matrix, and , then
In 2013, Cheng et al. [3] presented the following result: If and is a doubly stochastic matrix, then
In this paper, we present some new lower bounds of and , which improve (1) and (2).
2 Main results
In this section, we present our main results. Firstly, we give some lemmas.
Lemma 1 [4]
Let . If A is a strictly row diagonally dominant matrix, then satisfies
Lemma 2 Let . If A is a strictly row diagonally dominant M-matrix, then satisfies
where
Proof This proof is similar to the one of Lemma 2.2 in [3]. □
Lemma 3 If and is a doubly stochastic matrix, then
where is defined as in Lemma 2.
Proof This proof is similar to the one of Lemma 3.1 in [3]. □
Lemma 4 [4]
If is a strictly row diagonally dominant M-matrix, then, for ,
Lemma 5 [5]
If and are positive real numbers, then all the eigenvalues of A lie in the region
Lemma 6 [6]
If and are positive real numbers, then all the eigenvalues of A lie in the region
Theorem 1 If , and , then
where and is defined as in Lemma 2.
Proof It is evident that the result holds with equality for .
We next assume that .
Since A is an M-matrix, there exists a positive diagonal matrix D such that is a strictly row diagonally dominant M-matrix, and
Therefore, for convenience and without loss of generality, we assume that A is a strictly row diagonally dominant matrix.
(i) First, we assume that A and B are irreducible matrices. Then, for any , we have . Since is an eigenvalue of , then by Lemma 2 and Lemma 5, there exists an i such that
By Lemma 4, the above inequality and , for any , we obtain
(ii) Now, assume that one of A and B is reducible. It is well known that a matrix in is a nonsingular M-matrix if and only if all its leading principal minors are positive (see [7]). If we denote by the permutation matrix with , the remaining zero, then both and are irreducible nonsingular M-matrices for any chosen positive real number ϵ sufficiently small such that all the leading principal minors of both and are positive. Now, we substitute and for A and B, respectively, in the previous case, and then letting , the result follows by continuity. □
From Lemma 3 and Theorem 1, we can easily obtain the following corollaries.
Corollary 1 If and is a doubly stochastic matrix, then
Corollary 2 If and is a doubly stochastic matrix, then
Remark 1 We next give a simple comparison between (3) and (1), (4) and (2), respectively. Since , , , , then , and , for any , . Therefore,
So, the bound in (3) is bigger than the bound in (1) and the bound in (4) is bigger than the bound in (2).
Theorem 2 If and , then
where () is defined as in Theorem 1.
Proof It is evident that the result holds with equality for .
We next assume that . For convenience and without loss of generality, we assume that A is a strictly row diagonally dominant matrix.
(i)First, we assume that A and B are irreducible matrices. Let , , . Then, for any , , we have
Therefore, there exists a real number () such that
Hence,
Let . Obviously, (if , then A is reducible, which is a contradiction). Let
Since A is irreducible, then , , and . Let . By Lemma 6, there exist , such that
And by Lemma 2, we have
Therefore,
Furthermore, we obtain
that is,
(ii) Now, assume that one of A and B is reducible. We substitute and for A and B, respectively, in the previous case, and then letting , the result follows by continuity. □
Corollary 3 If and , then
Example 1 Let
It is easily proved that A and B are nonsingular M-matrices and A is a doubly stochastic matrix.
(i) If we apply Theorem 4.8 of [2], we have
If we apply Theorem 2.4 of [8], we have
But, if we apply Theorem 1, we have
If we apply Corollary 1, we have
If we apply Theorem 2, we have
-
(ii)
If we apply Theorem 3.2 of [3], we get
But, if we apply Corollary 2, we get
If we apply Corollary 3, we get
References
Fiedler M, Markham T: An inequality for the Hadamard product of an M -matrix and inverse M -matrix. Linear Algebra Appl. 1988, 101: 1–8.
Zhou DM, Chen GL, Wu GX, Zhang XY: On some new bounds for eigenvalues of the Hadamard product and the Fan product of matrices. Linear Algebra Appl. 2013, 438: 1415–1426. 10.1016/j.laa.2012.09.013
Cheng GH, Tan Q, Wang ZD: Some inequalities for the minimum eigenvalue of the Hadamard product of an M -matrix and its inverse. J. Inequal. Appl. 2013, 2013(65):1–9.
Yong XR, Wang Z: On a conjecture of Fiedler and Markham. Linear Algebra Appl. 1999, 288: 259–267.
Varga RS: Minimal Gerschgorin sets. Pac. J. Math. 1965, 15: 719–729. 10.2140/pjm.1965.15.719
Horn RA, Johnson CR: Matrix Analysis. Cambridge University Press, Cambridge; 1985.
Berman A, Plemmons RJ: Nonnegative Matrices in the Mathematical Sciences. SIAM, Philadelphia; 1994.
Zhou DM, Chen GL, Wu GX, Zhang XY: Some inequalities for the Hadamard product of an M -matrix and an inverse M -matrix. J. Inequal. Appl. 2013, 2013(16):1–10.
Acknowledgements
The authors are very indebted to the referees for their valuable comments and corrections, which improved the original manuscript of this paper. This work was supported by the National Natural Science Foundation of China (11361074), IRTSTYN and Foundation of Yunnan University (2012CG017).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to this work. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Li, Yt., Wang, F., Li, Cq. et al. Some new bounds for the minimum eigenvalue of the Hadamard product of an M-matrix and an inverse M-matrix. J Inequal Appl 2013, 480 (2013). https://doi.org/10.1186/1029-242X-2013-480
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1029-242X-2013-480