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New inequalities for the Hadamard product of an M-matrix and its inverse
Journal of Inequalities and Applications volume 2015, Article number: 35 (2015)
Abstract
For the Hadamard product \(A\circ A^{-1}\) of an M-matrix A and its inverse \(A^{-1}\), some new inequalities for the minimum eigenvalue of \(A\circ A^{-1}\) are derived. Numerical example is given to show that the inequalities are better than some known results.
1 Introduction
The set of all \(n\times n\) real matrices is denoted by \(\mathbb{R}^{n\times n}\), and \(\mathbb{C}^{n\times n}\) denotes the set of all \(n\times n\) complex matrices.
A matrix \(A=(a_{ij})\in\mathbb{R}^{n\times n}\) is called an M-matrix [1] if there exists a nonnegative matrix B and a nonnegative real number λ such that
where I is an identity matrix, \(\rho(B)\) is a spectral radius of the matrix B. If \(\lambda=\rho(B)\), then A is a singular M-matrix; if \(\lambda>\rho(B)\), then A is called a nonsingular M-matrix. Denote by \(M_{n}\) the set of all \(n\times n\) nonsingular M-matrices. Let us denote
and \(\sigma(A)\) denotes the spectrum of A. It is known that [2] \(\tau(A)=\frac{1}{\rho(A^{-1})}\) is a positive real eigenvalue of \(A\in M_{n}\).
The Hadamard product of two matrices \(A=(a_{ij})\) and \(B=(b_{ij})\) is the matrix \(A\circ B=(a_{ij}b_{ij})\). If A and B are M-matrices, then it is proved in [3] that \(A\circ B^{-1}\) is also an M-matrix.
A matrix A is irreducible if there does not exist any permutation matrix P such that
where \(A_{11}\) and \(A_{22}\) are square matrices.
For convenience, for any positive integer n, N denotes the set \(\{1,2,\ldots,n\}\). Let \(A=(a_{ij})\in\mathbb{R}^{n\times n}\) be a strictly diagonally dominant by row, for any \(i\in N\), denote
Recently, some lower bounds for the minimum eigenvalue of the Hadamard product of an M-matrix and its inverse have been proposed. Let \(A\in M_{n}\), it was proved in [4] that
Subsequently, Fiedler and Markham [3] gave a lower bound on \(\tau(A\circ A^{-1})\),
and conjectured that
Chen [5], Song [6] and Yong [7] have independently proved this conjecture.
In [8], Li et al. gave the following result:
Furthermore, if \(a_{11}=a_{22}=\cdots=a_{nn}\), they have obtained
In this paper, we present some new lower bounds for \(\tau(A\circ A^{-1})\). These bounds improve the results in [8–11].
2 Preliminaries and notations
In this section, we give some lemmas that involve inequalities for the entries of \(A^{-1}\). They will be useful in the following proofs.
Lemma 2.1
[7]
If \(A=(a_{ij})\in\mathbb{R}^{n\times n}\) is a strictly row diagonally dominant matrix, that is,
then \(A^{-1}=(b_{ij})\) exists, and
Lemma 2.2
Let \(A=(a_{ij})\in\mathbb{R}^{n\times n}\) be a strictly diagonally dominant M-matrix by row. Then, for \(A^{-1}=(b_{ij})\), we have
Proof
For \(i\in N\), let
and
Since A is strictly diagonally dominant, then \(0< d_{k}<1\) and \(0< s_{ji}<1\). Therefore, there exists \(\varepsilon>0\) such that \(0< d_{k}(\varepsilon)<1\) and \(0< s_{ji}(\varepsilon)<1\). For any \(i\in N\), let
Obviously, the matrix \(AS_{i}(\varepsilon)\) is also a strictly diagonally dominant M-matrix by row. Therefore, by Lemma 2.1, we derive the following inequality:
i.e.,
Let \(\varepsilon\longrightarrow0\) to obtain
This proof is completed. □
Lemma 2.3
Let \(A=(a_{ij})\in\mathbb{R}^{n\times n}\) be a strictly row diagonally dominant M-matrix. Then, for \(A^{-1}=(b_{ij})\), we have
Proof
Let \(B=A^{-1}\). Since A is an M-matrix, then \(B\geq0\). By \(AB=I\), we have
Hence
that is,
By Lemma 2.2, we have
i.e.,
Thus the proof is completed. □
Lemma 2.4
[12]
If \(A^{-1}\) is a doubly stochastic matrix, then \(Ae=e\), \(A^{T}e=e\), where \(e=(1,1,\ldots,1)^{T}\).
Lemma 2.5
[13]
Let \(A=(a_{ij})\in\mathbb{C}^{n\times n}\) and \(x_{1},x_{2},\ldots,x_{n}\) be positive real numbers. Then all the eigenvalues of A lie in the region
Lemma 2.6
[3]
If P is an irreducible M-matrix, and \(Pz\geq kz\) for a nonnegative nonzero vector z, then \(\tau(P)\geq k\).
3 Main results
In this section, we give two new lower bounds for \(\tau(A\circ A^{-1})\) which improve some previous results.
Theorem 3.1
Let \(A=(a_{ij})\in\mathbb{R}^{n\times n}\) be an M-matrix, and suppose that \(A^{-1}=(b_{ij})\) is doubly stochastic. Then
Proof
Since \(A^{-1}\) is doubly stochastic and A is an M-matrix, by Lemma 2.4, we have
and
The matrix A is strictly diagonally dominant by row. Then, by Lemma 2.2, for \(i\in N\), we have
i.e.,
This proof is completed. □
Theorem 3.2
Let \(A=(a_{ij})\in\mathbb{R}^{n\times n}\) be an M-matrix, and let \(A^{-1}=(b_{ij})\) be doubly stochastic. Then
Proof
It is evident that (3.1) is an equality for \(n=1\).
We next assume that \(n\geq2\).
Firstly, we assume that \(A^{-1}\) is irreducible. By Lemma 2.4, we have
and
Let
Since A is an irreducible matrix, then \(0< m_{j}\leq1\). Let \(\tau(A\circ A^{-1})=\lambda\), so that \(0<\lambda<a_{ii}b_{ii}\), \(i\in N\). Thus, by Lemma 2.5, there is a pair \((i,j)\) of positive integers with \(i\neq j\) such that
From inequality (3.2), we have
Thus, (3.3) is equivalent to
that is,
If A is reducible, without loss of generality, we may assume that A has the following block upper triangular form:
with irreducible diagonal blocks \(A_{ii}\), \(i=1,2,\ldots,s\). Obviously, \(\tau(A\circ A^{-1})=\min _{i}\tau(A_{ii}\circ A_{ii}^{-1})\). Thus, the problem of the reducible matrix A is reduced to those of irreducible diagonal blocks \(A_{ii}\). The result of Theorem 3.2 also holds. □
Theorem 3.3
Let \(A=(a_{ij})\in M_{n}\) and \(A^{-1}=b_{ij}\) be a doubly stochastic matrix. Then
Proof
Since \(A^{-1}\) is a doubly stochastic matrix, by Lemma 2.4, we have
For any \(j\neq i\), we have
or equivalently
So, we can obtain
and
Without loss of generality, for \(i\neq j\), assume that
Thus, (3.6) is equivalent to
Thus we have
This proof is completed. □
Remark 3.1
According to inequality (3.4), it is easy to know that
That is to say, the result of Lemma 2.2 is sharper than that of Theorem 2.1 in [8]. Moreover, the result of Theorem 3.2 is sharper than that of Theorem 3.1 in [8], respectively.
Theorem 3.4
Let \(A=(a_{ij})\in\mathbb{R}^{n\times n}\) be an irreducible strictly row diagonally dominant M-matrix. Then
Proof
Since A is irreducible, then \(A^{-1}>0\), and \(A\circ A^{-1}\) is again irreducible. Note that
Let
where \(e=(1,1,\ldots,1)^{T}\). Without loss of generality, we may assume that \(t_{1}=\min _{i} \{t_{i} \}\), by Lemma 2.2, we have
Therefore, by Lemma 2.6, we have
This proof is completed. □
Remark 3.2
According to inequality (3.5), we can get
That is to say, the bound of Theorem 3.4 is sharper than the bound of Theorem 3.5 in [8].
Remark 3.3
If A is an M-matrix, we know that there exists a diagonal matrix D with positive diagonal entries such that \(D^{-1}AD\) is a strictly row diagonally dominant M-matrix. So the result of Theorem 3.4 also holds for a general M-matrix.
4 Example
Consider the following M-matrix:
Since \(Ae=e\) and \(A^{T}e=e\), \(A^{-1}\) is doubly stochastic. By calculations we have
(1) Estimate the upper bounds for entries of \(A^{-1}=(b_{ij})\) . If we apply Theorem 2.1(a) of [8], we have
If we apply Lemma 2.2, we have
Combining the result of Lemma 2.2 with the result of Theorem 2.1(a) of [8], we see that the result of Lemma 2.2 is the best.
By Theorem 2.3 and Lemma 3.2 of [8], we can get the following bounds for the diagonal entries of \(A^{-1}\):
By Lemma 2.3 and Theorem 3.1, we obtain
(2) Lower bounds for \(\tau(A\circ A^{-1})\).
By the conjecture of Fiedler and Markham, we have
By Theorem 3.1 of [8], we have
By Corollary 2.5 of [9], we have
By Theorem 3.1 of [10], we have
By Corollary 2 of [11], we have
If we apply Theorem 3.2, we have
The numerical example shows that the bound of Theorem 3.2 is better than these corresponding bounds in [8–11].
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Acknowledgements
The author is grateful to the referees for their useful and constructive suggestions. This research is supported by the Scientific Research Fund of Yunnan Provincial Education Department (2013C165).
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Chen, Fb. New inequalities for the Hadamard product of an M-matrix and its inverse. J Inequal Appl 2015, 35 (2015). https://doi.org/10.1186/s13660-015-0555-1
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DOI: https://doi.org/10.1186/s13660-015-0555-1