Some inequalities for the minimum eigenvalue of the Hadamard product of an M-matrix and its inverse
© Cheng et al.; licensee Springer 2013
Received: 31 July 2012
Accepted: 24 January 2013
Published: 21 February 2013
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.
KeywordsHadamard product M-matrix inverse M-matrix strictly diagonally dominant matrix eigenvalue
if , where , .
For any two matrices and , the Hadamard product of A and B is . If , then is also an M-matrix .
where and are square matrices.
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 
This proof is completed. □
Thus the proof is completed. □
Lemma 2.4 
Lemma 2.5 
If is a doubly stochastic matrix, then , , where .
3 Main results
where is an irreducible diagonal block matrix, . Obviously, . Thus the reducible case is converted into the irreducible case. This proof is completed. □
This proof is completed. □
That is to say, the result of Lemma 2.2 is sharper than the ones of Theorem 2.1 in  and Lemma 2.2 in . Moreover, the results of Theorem 3.2 and Theorem 3.3 are sharper than the ones of Theorem 3.1 in  and Theorem 3.3 in , respectively.
Proof This proof is similar to the one of Theorem 3.5 in . □
Remark 3.3 Using the above similar ideas, we can obtain similar inequalities of the strictly diagonally M-matrix by column.
- 1.Estimate the upper bounds for entries of . Firstly, by Lemma 2.2(2) in , we have
Lower bounds for .
This research is supported by National Natural Science Foundations of China (No. 11101069).
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