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
where
and
is a positive real eigenvalue of
if
For any two
A matrix A is irreducible if there does not exist a permutation matrix P such that
where
For convenience, the set
Recently, some lower bounds for the minimum eigenvalue of the Hadamard product of
an Mmatrix and an inverse Mmatrix have been proposed. Let
In [9], Li et al. gave the following result:
Furthermore, if
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
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]
Let
If
Lemma 2.2Let
Proof Firstly, we consider
and
Since A is strictly diagonally dominant, then
Similarly to the proofs of Theorem 2.1 and Theorem 2.4 in [8], we can prove that the matrix
i.e.,
Let
This proof is completed. □
Lemma 2.3Let
Proof Let
Hence
or equivalently,
Furthermore, by Lemma 2.2, we get
i.e.,
Thus the proof is completed. □
Lemma 2.4[10]
Let
Lemma 2.5[11]
If
3 Main results
In this section, we give two new lower bounds for
Lemma 3.1If
Proof This proof is similar to the ones of Lemma 3.2 in [8] and Theorem 3.2 in [9]. □
Theorem 3.1Let
Proof Firstly, we assume that A is irreducible. By Lemma 2.5, we have
Denote
Since A is an irreducible matrix, we know 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
Theorem 3.2If
Proof Since A is strictly diagonally dominant by row, for any
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 3.3If
Proof Since A is strictly diagonally dominant by row, for any
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 3.4If
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
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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