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1 Introduction
Let \(C^{m\times n}\) \((R^{m\times n})\) be the set of all complex (real) matrices and let \(\mathbb{M}_{n}^{+}\) be the positive definite Hermitian matrices. Let \(Z^{n\times n}=\{A=(a_{ij})\in R^{n\times n}:a_{ij} \leq0, i\neq j, i, j \in\{1,2,\ldots,n\}\}\). For any \(A=(a_{ij}) \in C^{n\times n}\), its associated matrix is defined by \(A^{\prime}=( \alpha_{ij})\), where \(\alpha_{ii}=\vert a_{ii}\vert \), \(\alpha_{ij}=-\vert a_{ij}\vert \) (\(i\neq j\)). For \(A=(a_{ij})\), \(B=(b_{ij})\) \(\in C^{m\times n}\), the Hadamard product of A and B is \(A \circ B =(a_{ij}b_{ij})\in C ^{m\times n}\) while their Fan product \(A*B=(c_{ij})\) is defined by \(c_{ii}=a_{ii}b_{ii}\) and \(c_{ij}=-a_{ij}b_{ij}\) for \(i\neq j\).
If \(A=(a_{ij}) \in C^{n\times n}\), then the \(k \times k\) leading principal submatrix of A is denoted by \(A_{k}\) (\(k\in\{1,2,\ldots,n\}\)). \(A_{\alpha}\) denotes the principal submatrix of A, with indices in \(\alpha\subseteq\{1,2,\ldots,n\}\). \(A\in R^{n\times n}\) is called an M-matrix if \(A\in Z^{n\times n}\) and \(\det A_{k}>0\) (\(\forall k\in\{1,2,\ldots,n\}\)), and we denote it by \(A\in M_{n}\). A matrix \(A\in C^{n\times n}\) is called an H-matrix if \(A^{\prime}\) is an M-matrix, and we denote it by \(A\in H_{n}\).
Lynn [1], Theorem 3.1, proved the following determinantal inequality for H-matrices: if \(A, B\in H_{n}\), then
i.e.
Chen [2], Theorem 2.7, obtained a determinantal inequality for positive definite matrices: if \(A=(a_{ij})\), \(B=(b_{ij})\in \mathbb{M}_{n}^{+}\), then
Lin [3] recently proved that a similar result to the block positive definite matrices holds for the block Hadamard product.
Ando [4], Theorem 5.3, has given the following result: if \(A=(a_{ij})\), \(B=(b_{ij})\) are M-matrices, then
i.e.
In this paper, we will present some determinantal inequalities for matrices which are generalizations of (1.1), (1.2), and (1.3).
2 Main results and some remarks
We give some lemmas before we present the main theorems of this paper.
Lemma 1
[4], Corollary 4.1.2
Let \(A=(a_{ij})\in R^{n\times n}\) be an M-matrix. If \(\alpha_{i}\subseteq\{1,2,\ldots,n\}\) (\(i=1,2,3, \ldots,N\)) satisfies \(\alpha_{i}\cap\alpha_{j}=\phi\) for \(i\neq j\) and \(\bigcup_{j=1}\alpha_{j}=\{1,2,\ldots,n\}\), then
In particular,
Lemma 2
[1], Theorem 3.1
If A, B are H-matrices and \(C=A\circ B\), then C is H-matrix.
Lemma 3
[5], Theorem 5.2.1
If A, B are positive definite matrices and \(C=A\circ B\), then C is positive definite matrix.
Lemma 4
[6]
If A, B are M-matrices and \(C=A* B\), then C is M-matrix.
Now we present the main results.
First of all, we give a determinantal inequality for the Hadamard product of finite number of H-matrices as follows:
Theorem 5
If \(A_{1}=(a_{1}^{kl}), A_{2}=(a_{2}^{kl}),\ldots, A_{m}=(a_{m}^{kl})\) (\(k,l=1,\ldots,n\)) are H-matrices, then
Proof
By Lemma 2, it is straightforward to observe that the Hadamard product \(A_{1}\circ\cdots\circ A_{m}\) is an H-matrix. Use induction on k. When \(k=2\), the result is (1.1). Suppose that (2.2) holds when \(k=m-1\)
When \(k=m\), we need to show
By (1.1), we have
By the inductive assumption, the above inequality is
Let
By (2.1), we have
and so
Thus by \(ab\ge a+b-1\) for \(a, b\ge1\), the above inequality (2.3) is
This completes the proof. □
Remark 6
The above inequality in Theorem 5 is a generalization of the inequality (1.1).
Second, we achieve a determinantal inequality for the Hadamard product of positive definite matrices as follows:
Theorem 7
If \(A_{i}\) (\(i=1,\ldots,m\)) (\(m\ge2\)) are \(n\times n\) positive definite matrices, the Hadamard product of \(A_{i}=(a_{i}^{lt})\) and \(A_{j}=(a_{j}^{lt}) \) (\(i\neq j\)) is denoted by \(A_{i}\circ A_{j}\), and \(A_{i}^{(k)}\) is the \(k\times k\) (\(k=1,2,\ldots,n\)) leading principal submatrix of \(A_{i}\), then
Proof
By Lemma 3, it is straightforward to see that the Hadamard product \(A_{1}\circ\cdots\circ A_{m}\) is a positive definite matrix. Use induction on m. When \(k=2\), the result is (1.2). Suppose that (2.4) holds when \(k=m-1\). We have
When \(k=m\), we need to show
By (1.2), we have
By the inductive assumption, the above inequality is such that
Let
By Fischer’s inequality [5], p.506, we have
and so
Thus by \(a_{\mu}b_{\mu}\ge a_{\mu}+b_{\mu}-1\) for \(a_{\mu}, b _{\mu}\ge1\), the above inequality (2.5) is
This completes the proof. □
Remark 8
The inequality in Theorem 7 is a generalization of the inequality (1.2).
Finally, a result on Fan product of M-matrices is obtained in the following theorem.
Theorem 9
If \(A_{1}=(a_{1}^{kl}), A_{2}=(a_{2}^{kl}),\ldots, A_{m}=(a_{m}^{kl})\) (\(k,l=1,\ldots,n\)) are M-matrices, then
Proof
By Lemma 4, it is straightforward to see that the Hadamard product \(A_{1}\ast\cdots\ast A_{m}\) is an M-matrix. Use induction on k. When \(k=2\), the result is (1.3). Let \(k=m-1\), (2.6) holds:
When \(k=m\), we need to show
By (1.3), we have
By the inductive assumption, the above inequality is
Let
By (2.1), we have
and so
So by \(ab\ge a+b-1\) for \(a, b\ge1\), the above inequality (2.7) is
This completes the proof. □
Remark 10
The inequality in Theorem 9 is a generalization of the inequality (1.3).
References
Lynn, MS: On the Schur product of the H-matrices and non-negative matrices and related inequalities. Proc. Camb. Philos. 60, 425-431 (1964)
Chen, S: Some determinantal inequalities for Hadamard product of matrices. Linear Algebra Appl. 368, 99-106 (2003)
Lin, M: An Oppenheim type inequality for a block Hadamard product. Linear Algebra Appl. 452, 1-6 (2014)
Ando, T: Inequalities for M-matrices. Linear Multilinear Algebra 8, 291-316 (1980)
Horn, RA, Johnson, CR: Matrix Analysis, 2nd edn. Cambridge University Press, Cambridge (2013)
Fan, K: Note on M-matrices. Indag. Math. 67, 43-49 (1964)
Acknowledgements
We are grateful to Dr. Limin Zou for fruitful discussions. This research was supported by the key project of the applied mathematics of Hainan Normal University, the natural science foundation of Hainan Province (No. 20161005), the Chongqing Graduate Student Research Innovation Project (No. CYS14020) and the Doctoral scientific research foundation of Hainan Normal University.
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Authors’ contributions
Xiaohui Fu carried out all the proofs of the results and gave the generalizations of Fan product. Yang Liu participated in the design of the study and drafted the manuscript. All authors read and approved the final manuscript.
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Fu, X., Liu, Y. Some determinantal inequalities for Hadamard and Fan products of matrices. J Inequal Appl 2016, 262 (2016). https://doi.org/10.1186/s13660-016-1208-8
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DOI: https://doi.org/10.1186/s13660-016-1208-8