Some inequalities for the minimum eigenvalue of the Hadamard product of an M-matrix and its inverse

Guanghui Cheng*, Qin Tan and Zhuande Wang

Author Affiliations

School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan, 611731, P.R. China

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Journal of Inequalities and Applications 2013, 2013:65  doi:10.1186/1029-242X-2013-65

 Received: 31 July 2012 Accepted: 24 January 2013 Published: 21 February 2013

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, some new inequalities for the minimum eigenvalue of the Hadamard product of an M-matrix and its inverse are given. These inequalities are sharper than the well-known results. A simple example is shown.

AMS Subject Classification: 15A18, 15A42.

Keywords:
Hadamard product; M-matrix; inverse M-matrix; strictly diagonally dominant matrix; eigenvalue

1 Introduction

A matrix is called a nonnegative matrix if . A matrix is called a nonsingular M-matrix [1] if there exist and such that

where is a spectral radius of the nonnegative matrix B, is the identity matrix. Denote by the set of all nonsingular M-matrices. The matrices in are called inverse M-matrices. Let us denote

and denotes the spectrum of A. It is known that [2]

is a positive real eigenvalue of and the corresponding eigenvector is nonnegative. Indeed

if , where , .

For any two matrices and , the Hadamard product of A and B is . If , then is also an M-matrix [3].

A matrix A is irreducible if there does not exist a permutation matrix P such that

where and are square matrices.

For convenience, the set is denoted by N, where n (≥3) is any positive integer. Let be a strictly diagonally dominant by row, denote

Recently, some lower bounds for the minimum eigenvalue of the Hadamard product of an M-matrix and an inverse M-matrix have been proposed. Let , for example, has been proven by Fiedler et al. in [4]. Subsequently, was given by Fiedler and Markham in [3], and they conjectured that . Song [5], Yong [6] and Chen [7] have independently proven this conjecture. In [8], Li et al. improved the conjecture when is a doubly stochastic matrix and gave the following result:

In [9], Li et al. gave the following result:

Furthermore, if , they have obtained

i.e., under this condition, the bound of [9] is better than the one of [8].

In this paper, our motives are to improve the lower bounds for the minimum eigenvalue . The main ideas are based on the ones of [8] and [9].

2 Some preliminaries and notations

In this section, we give some notations and lemmas which mainly focus on some inequalities for the entries of the inverse M-matrix and the strictly diagonally dominant matrix.

Lemma 2.1[6]

Letbe a strictly diagonally dominant matrix by row, i.e.,

If, then

Lemma 2.2Letbe a strictly diagonally dominantM-matrix by row. If, then

Proof Firstly, we consider is a strictly diagonally dominant M-matrix by row. For , let

and

Since A is strictly diagonally dominant, then and . Therefore, there exists such that and . Let us define one positive diagonal matrix

Similarly to the proofs of Theorem 2.1 and Theorem 2.4 in [8], we can prove that the matrix is also a strictly diagonally dominant M-matrix by row for any . Furthermore, by Lemma 2.1, we can obtain the following result:

i.e.,

Let to get

This proof is completed. □

Lemma 2.3Letbe a strictly diagonally dominant matrix by row and, then we have

Proof Let . Since A is an M-matrix, then . By , we have

Hence

or equivalently,

Furthermore, by Lemma 2.2, we get

i.e.,

Thus the proof is completed. □

Lemma 2.4[10]

Letandbe positive real numbers. Then all the eigenvalues ofAlie in the region

Lemma 2.5[11]

Ifis a doubly stochastic matrix, then, , where.

3 Main results

In this section, we give two new lower bounds for which improve the ones in [8] and [9].

Lemma 3.1Ifandis a doubly stochastic matrix, then

Proof This proof is similar to the ones of Lemma 3.2 in [8] and Theorem 3.2 in [9]. □

Theorem 3.1Letandbe a doubly stochastic matrix. Then

Proof Firstly, we assume that A is irreducible. By Lemma 2.5, we have

Denote

Since A is an irreducible matrix, we know that . So, by Lemma 2.4, there exists such that

or equivalently,

Secondly, if A is reducible, without loss of generality, we may assume that A has the following block upper triangular form:

where is an irreducible diagonal block matrix, . Obviously, . Thus the reducible case is converted into the irreducible case. This proof is completed. □

Theorem 3.2Ifis a strictly diagonally dominant by row, then

Proof Since A is strictly diagonally dominant by row, for any , we have

or equivalently,

(1)

So, we can obtain

(2)

and

Therefore, it is easy to obtain that

Obviously, we have the desired result

This proof is completed. □

Theorem 3.3Ifis strictly diagonally dominant by row, then

Proof Since A is strictly diagonally dominant by row, for any , we have

i.e.,

(3)

So, we can obtain

(4)

and

Therefore, it is easy to obtain that

Obviously, we have the desired result

□

Remark 3.1 According to inequalities (1) and (3), it is easy to know that

and

That is to say, the result of Lemma 2.2 is sharper than the ones of Theorem 2.1 in [8] and Lemma 2.2 in [9]. Moreover, the results of Theorem 3.2 and Theorem 3.3 are sharper than the ones of Theorem 3.1 in [8] and Theorem 3.3 in [9], respectively.

Theorem 3.4Ifis strictly diagonally dominant by row, then

Proof This proof is similar to the one of Theorem 3.5 in [8]. □

Remark 3.2 According to inequalities (2) and (4), we get

and

That is to say, the bound of Theorem 3.4 is sharper than the ones of Theorem 3.5 in [8] and Theorem 3.4 in [9], respectively.

Remark 3.3 Using the above similar ideas, we can obtain similar inequalities of the strictly diagonally M-matrix by column.

4 Example

For convenience, we consider the M-matrix A is the same as the matrix of [8]. Define the M-matrix A as follows:

1. Estimate the upper bounds for entries of . Firstly, by Lemma 2.2(2) in [9], we have

By Lemma 2.2, we have

By Lemma 2.3 and Theorem 3.1 in [9], we get

By Lemma 2.3 and Lemma 3.1, we get

2. Lower bounds for .

By Theorem 3.2 in [9], we obtain

By Theorem 3.1, we obtain

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.

Acknowledgements

This research is supported by National Natural Science Foundations of China (No. 11101069).

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