Journal of Inequalities and Applications

Impact Factor 0.773

Open Access

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

Journal of Inequalities and Applications20132013:65

DOI: 10.1186/1029-242X-2013-65

Accepted: 24 January 2013

Published: 21 February 2013

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]

Let be a strictly diagonally dominant matrix by row, i.e.,
If , then
Lemma 2.2 Let be a strictly diagonally dominant M-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 2.3 Let be 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]

Let and be positive real numbers. Then all the eigenvalues of A lie in the region

Lemma 2.5 [11]

If is 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 3.1 If and is 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 3.1 Let and be 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 3.2 If is 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 3.3 If is 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 3.4 If is 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. 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
1. 2.

Lower bounds for .

By Theorem 3.2 in [9], we obtain
By Theorem 3.1, we obtain

Declarations

Acknowledgements

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

Authors’ Affiliations

(1)
School of Mathematical Sciences, University of Electronic Science and Technology of China

References

1. Berman A, Plemmons RJ Classics in Applied Mathematics 9. In Nonnegative Matrices in the Mathematical Sciences. SIAM, Philadelphia; 1994.View Article
2. Horn RA, Johnson CR: Topics in Matrix Analysis. Cambridge University Press, Cambridge; 1991.
3. Fiedler M, Markham TL: An inequality for the Hadamard product of an M -matrix and inverse M -matrix. Linear Algebra Appl. 1988, 101: 1–8.
4. Fiedler M, Johnson CR, Markham T, Neumann M: A trace inequality for M -matrices and the symmetrizability of a real matrix by a positive diagonal matrix. Linear Algebra Appl. 1985, 71: 81–94.
5. Song YZ: On an inequality for the Hadamard product of an M -matrix and its inverse. Linear Algebra Appl. 2000, 305: 99–105. 10.1016/S0024-3795(99)00224-4
6. Yong XR: Proof of a conjecture of Fiedler and Markham. Linear Algebra Appl. 2000, 320: 167–171. 10.1016/S0024-3795(00)00211-1
7. Chen SC: A lower bound for the minimum eigenvalue of the Hadamard product of matrix. Linear Algebra Appl. 2004, 378: 159–166.
8. Li HB, Huang TZ, Shen SQ, Li H: Lower bounds for the eigenvalue of Hadamard product of an M -matrix and its inverse. Linear Algebra Appl. 2007, 420: 235–247. 10.1016/j.laa.2006.07.008
9. Li YT, Chen FB, Wang DF: New lower bounds on eigenvalue of the Hadamard product of an M -matrix and its inverse. Linear Algebra Appl. 2009, 430: 1423–1431. 10.1016/j.laa.2008.11.002
10. Varga RS: Minimal Gerschgorin sets. Pac. J. Math. 1965, 15(2):719–729. 10.2140/pjm.1965.15.719
11. Sinkhorn R: A relationship between arbitrary positive matrices and doubly stochastic matrices. Ann. Math. Stat. 1964, 35: 876–879. 10.1214/aoms/1177703591