Abstract
In this paper, some new inequalities for the minimum eigenvalue of the Hadamard product of an Mmatrix and its inverse are given. These inequalities are sharper than the wellknown results. A simple example is shown.
AMS Subject Classification: 15A18, 15A42.
Keywords:
Hadamard product; Mmatrix; inverse Mmatrix; strictly diagonally dominant matrix; eigenvalue1 Introduction
A matrix is called a nonnegative matrix if . A matrix is called a nonsingular Mmatrix [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 Mmatrices. The matrices in are called inverse Mmatrices. 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
For any two matrices and , the Hadamard product of A and B is . If , then is also an Mmatrix [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 Mmatrix and an inverse Mmatrix 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 Mmatrix and the strictly diagonally dominant matrix.
Lemma 2.1[6]
Letbe a strictly diagonally dominant matrix by row, i.e.,
Lemma 2.2Letbe a strictly diagonally dominantMmatrix by row. If, then
Proof Firstly, we consider is a strictly diagonally dominant Mmatrix 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 Mmatrix by row for any . Furthermore, by Lemma 2.1, we can obtain the following result:
i.e.,
This proof is completed. □
Lemma 2.3Letbe a strictly diagonally dominant matrix by row and, then we have
Proof Let . Since A is an Mmatrix, 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]
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,
So, we can obtain
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.,
So, we can obtain
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 Mmatrix by column.
4 Example
For convenience, we consider the Mmatrix A is the same as the matrix of [8]. Define the Mmatrix 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
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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