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New inequalities for the Hadamard product of an M-matrix and an inverse M-matrix
Journal of Inequalities and Applications volume 2015, Article number: 368 (2015)
Abstract
Let A and B be nonsingular M-matrices. Some new convergent sequences of the lower bounds of the minimum eigenvalue \(\tau(B\circ A^{-1})\) for the Hadamard product of B and \(A^{-1}\) are given. Numerical examples are given to show that these sequences are better than some known results and could reach the true value of the minimum eigenvalue in some cases.
1 Introduction
For a positive integer n, N denotes the set \(\{1, 2, \ldots, n\}\), and \(\mathbb{R}^{n\times n}(\mathbb{C}^{n\times n})\) denotes the set of all \({n\times n}\) real (complex) matrices throughout.
A matrix \(A=[a_{ij}]\in\mathbb{R}^{n\times n}\) is called a nonsingular M-matrix if \(a_{ij}\leq0\), \(i,j\in N\), \(i\neq j\), AÂ is nonsingular and \(A^{-1}\geq0\) (see [1]). Denote by \(M_{n}\) the set of all \(n\times n\) nonsingular M-matrices.
If A is a nonsingular M-matrix, then there exists a positive eigenvalue of A equal to \(\tau(A)\equiv[\rho(A^{-1})]^{-1}\), where \(\rho(A^{-1})\) is the Perron eigenvalue of the nonnegative matrix \(A^{-1}\). It is easy to prove that \(\tau(A)=\min\{|\lambda|:\lambda\in\sigma(A)\}\), where \(\sigma(A)\) denotes the spectrum of A (see [2]).
A matrix A is called reducible if there exists a nonempty proper subset \(I\subset N\) such that \(a_{ij}=0\), \(\forall i \in I\), \(\forall j\notin I\). If A is not reducible, then we call A irreducible (see [3]).
For two real matrices \(A=[a_{ij}]\) and \(B=[b_{ij}]\) of the same size, the Hadamard product of A and B is defined as the matrix \(A\circ B=[a_{ij}b_{ij}]\). If A and B are two nonsingular M-matrices, then it was proved in [4] that \(A\circ B^{-1}\) is also a nonsingular M-matrix.
Let \(A=[a_{ij}]\in M_{n}\). For \(i,j,k\in N\), \(j\neq i\), denote
In 2015, Chen [5] gave the following result: Let \(A=[a_{ij}]\in M_{n}\) and \(A^{-1}=[\alpha_{ij}]\) be a doubly stochastic matrix. Then
where
Soon after, Zhao et al. [6] obtained the following result: Let \(A=[a_{ij}],B=[b_{ij}]\in M_{n}\). Then, for \(t=1,2,\ldots\) ,
where
In this paper, we present some new convergent sequences of the lower bounds of \(\tau(B\circ A^{-1})\) and \(\tau(A\circ A^{-1})\), which improve (1) and (2). Numerical examples show that these sequences could reach the true value of \(\tau(A\circ A^{-1})\) in some cases.
2 Some lemmas
In this section, we give the following lemmas. These will be useful in the following proofs.
Lemma 1
[6]
If \(A=[a_{ij}]\in M_{n}\) is strictly row diagonally dominant, then \(A^{-1}=[\alpha_{ij}]\) exists, and for all \(i,j\in{N}\), \(j\neq{i}\), \(t=1,2,\ldots\) ,
Lemma 2
[6]
If \(A\in M_{n}\) and \(A^{-1}=[\alpha_{ij}]\) is a doubly stochastic matrix, then
Lemma 3
[7]
If \(A^{-1}\) is a doubly stochastic matrix, then \(A^{T}e=e\), \(Ae=e\), where \(e=(1, 1, \ldots, 1)^{T}\).
Lemma 4
[8]
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
3 Main results
In this section, we give several convergent sequences for \(\tau(B\circ{A^{-1}})\) and \(\tau(A\circ{A^{-1}})\).
Theorem 1
Let \(A=[a_{ij}],B=[b_{ij}]\in M_{n}\) and \(A^{-1}=[\alpha_{ij}]\). Then, for \(t=1,2,\ldots\) ,
Proof
It is evident that the result holds with equality for \(n=1\).
We next assume that \(n\geq2\).
Since \(A\in M_{n}\), there exists a positive diagonal matrix D such that \(D^{-1}AD\) is a strictly row diagonally dominant M-matrix, and
Therefore, for convenience and without loss of generality, we assume that A is a strictly row diagonally dominant matrix.
(a) First, we assume that A and B are irreducible matrices. Since A is irreducible, \(0< p_{i}^{(t)}<1\), for any \(i\in N\). Let \(\tau(B\circ A^{-1})=\lambda\). Since λ is an eigenvalue of \(B\circ A^{-1}\), \(0<\lambda< b_{ii}\alpha_{ii}\). By Lemma 1 and Lemma 4, there is a pair \((i,j)\) of positive integers with \(i\neq j\) such that
From inequality (4), we have
Thus, (5) is equivalent to
That is,
(b) Now, assume that one of A and B is reducible. It is well known that a matrix in \(Z_{n}=\{A=[a_{ij}]\in\mathbb{R}^{n\times n}:a_{ij}\leq0,i\neq{j}\}\) is a nonsingular M-matrix if and only if all its leading principal minors are positive (see condition (E17) of Theorem 6.2.3 of [1]). If we denote by \(C=[c_{ij}]\) the \(n\times n\) permutation matrix with \(c_{12}=c_{23}=\cdots=c_{n-1,n}=c_{n1}=1\), the remaining \(c_{ij}\) zero, then both \(A-{\varepsilon}C\) and \(B-{\varepsilon}C\) are irreducible nonsingular M-matrices for any chosen positive real number ε, sufficiently small such that all the leading principal minors of both \(A-{\varepsilon} C\) and \(B-{\varepsilon}C\) are positive. Now we substitute \(A-{\varepsilon} C\) and \(B-{\varepsilon}C\) for A and B, in the previous case, and then letting \({\varepsilon}\rightarrow0\), the result follows by continuity. □
Theorem 2
The sequence \(\{\Omega_{t}\}\), \(t=1,2,\ldots\) obtained from Theorem 1 is monotone increasing with an upper bound \(\tau(B\circ A^{-1})\) and, consequently, is convergent.
Proof
By Lemma 1, we have \(p^{(t)}_{ji}\geq p^{(t+1)}_{ji}\geq 0\), \(t=1,2,\ldots\) , so by the definition of \(p^{(t)}_{i}\), it is easy to see that the sequence \(\{p^{(t)}_{i}\}\) is monotone decreasing. Then \(\Omega_{t}\) is a monotonically increasing sequence. Hence, the sequence is convergent. □
Next, we give the following comparison theorem for (2) and (3).
Theorem 3
Let \(A=[a_{ij}],B=[b_{ij}]\in M_{n}\) and \(A^{-1}=[\alpha_{ij}]\). Then, for \(t=1,2,\ldots\) ,
Proof
Without loss of generality, for any \(i\neq j\), assume that
Thus, (6) is equivalent to
From (6), (7), and Lemma 1, we have
Thus we have
This proof is completed. □
Using Lemma 2 in (8), it can be seen that the following corollary holds clearly.
Corollary 1
Let \(A=[a_{ij}],B=[b_{ij}]\in M_{n}\) and \(A^{-1}=[\alpha_{ij}]\) be a doubly stochastic matrix. Then, for \(t=1,2,\ldots\) ,
Remark 1
Theorem 3 and Corollary 1 show that the bound in (3) is bigger than the bound in (2) and the bound in Corollary 1 of [6].
If \(B=A\), according to Theorem 1 and Corollary 1, the following corollaries are established, respectively.
Corollary 2
Let \(A=[a_{ij}]\in M_{n}\) and \(A^{-1}=[\alpha_{ij}]\). Then, for \(t=1,2,\ldots\) ,
Corollary 3
Let \(A=[a_{ij}]\in M_{n}\) and \(A^{-1}=[\alpha _{ij}]\) be a doubly stochastic matrix. Then, for \(t=1,2,\ldots\) ,
Remark 2
(a) We give a simple comparison between (1) and (9). According to Lemma 1, we know that \(s_{ji}\geq q_{ji}=\min\{{s_{ji},m_{ji}}\}\) and \(1\geq h_{i}\geq0\), so it is easy to see that \(u_{ji}\geq{v^{(0)}_{ji}}\geq p^{(t)}_{ji}\). Furthermore, by the definition of \(u_{i}\), \(p^{(t)}_{i}\), we have \(u_{i}\geq p^{(t)}_{i}\). Obviously, for \(t=1,2,\ldots\) , the bound in (9) is bigger than the bound in (1).
(b) Corollary 3 shows that the bound in Corollary 2 is bigger than the bound in Corollary 2 of [6].
Similar to the proof of Theorem 1, Theorem 2 and Theorem 3, we can obtain Theorem 4, Theorem 5, and Theorem 6, respectively.
Theorem 4
Let \(A=[a_{ij}],B=[b_{ij}]\in M_{n}\) and \(A^{-1}=[\alpha_{ij}]\). Then, for \(t=1,2,\ldots\) ,
Theorem 5
The sequence \(\{\Delta_{t}\}\), \(t=1,2,\ldots\) obtained from Theorem 4 is monotone increasing with an upper bound \(\tau(B\circ A^{-1})\) and, consequently, is convergent.
Theorem 6
Let \(A=[a_{ij}],B=[b_{ij}]\in M_{n}\) and \(A^{-1}=[\alpha_{ij}]\). Then, for \(t=1,2,\ldots\) ,
Corollary 4
Let \(A=[a_{ij}],B=[b_{ij}]\in M_{n}\) and \(A^{-1}\) be a doubly stochastic matrix. Then, for \(t=1,2,\ldots\) ,
Remark 3
Theorem 6 and Corollary 4 show that the bound in Theorem 4 is bigger than the bound in Theorem 3 of [6] and the bound in Corollary 3 of [6].
If \(B=A\), according to Theorem 4 and Corollary 4, the following corollaries are established, respectively.
Corollary 5
Let \(A=[a_{ij}]\in M_{n}\) and \(A^{-1}=[\alpha_{ij}]\). Then, for \(t=1,2,\ldots\) ,
Corollary 6
Let \(A=[a_{ij}]\in M_{n}\) and \(A^{-1}\) be a doubly stochastic matrix. Then, for \(t=1,2,\ldots\) ,
Remark 4
Corollary 6 shows that the bound in Corollary 5 is bigger than the bound in Corollary 4 of [6].
Let \(\Upsilon_{t}=\max\{\Gamma_{t},\mathrm{T}_{t}\}\). By Corollary 2 and Corollary 5, the following theorem is easily found.
Theorem 7
Let \(A=[a_{ij}]\in M_{n}\) and \(A^{-1}\) be a doubly stochastic matrix. Then, for \(t=1,2,\ldots\) ,
4 Numerical examples
In this section, several numerical examples are given to verify the theoretical results.
Example 1
Let
Based on \(A\in Z_{n}\) and \(Ae=e\), \(A^{T}e=e\), it is easy to see that A is nonsingular M-matrix and \(A^{-1}\) is doubly stochastic. Numerical results are given in Table 1 for the total number of iterations \(T=10\). In fact, \(\tau(A\circ{A^{-1}})=0.9678\).
Remark 5
The numerical results in Table 1 show that:
-
(a)
The lower bounds obtained from Theorem 7 are bigger than these corresponding bounds in [4–6, 9, 10].
-
(b)
The sequence obtained from Theorem 7 is monotone increasing.
-
(c)
The sequence obtained from Theorem 7 approximates effectively the true value of \(\tau(A\circ A^{-1})\), so we can estimate \(\tau(A\circ A^{-1})\) by Theorem 7.
Example 2
Let \(A=[a_{ij}]\in\mathbb{R}^{n\times n}\), where \(a_{11}=a_{22}=\cdots=a_{n,n}=2\), \(a_{12}=a_{23}=\cdots=a_{n-1,n}=a_{n,1}=-1\), and \(a_{ij}=0\) elsewhere.
It is easy to see that A is a nonsingular M-matrix and \(A^{-1}\) is doubly stochastic. If we apply Theorem 7 for \(n=10\) and \(n=100\), we have \(\tau(A\circ{A^{-1}})=0.7507\) and \(\tau(A\circ{A^{-1}})=0.7500\) when \(t=1\), respectively. In fact, \(\tau(A\circ{A^{-1}})=0.7507\) for \(n=10\) and \(\tau(A\circ{A^{-1}})=0.7500\) for \(n=100\).
Remark 6
Numerical results in Example 2 show that the lower bound obtained from Theorem 7 could reach the true value of \(\tau(A\circ A^{-1})\) in some cases.
5 Further work
In this paper, we present a new convergent sequence \(\{\Upsilon_{t}\}\), \(t=1,2,\ldots\) , which is more accurate than the convergent sequence in Theorem 5 of [6], to approximate \(\tau(A\circ A^{-1})\), and we do not give the error analysis, i.e., how accurately these bounds can be computed. At present, it is very difficult for the authors to do this. Next, we will study this problem.
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Acknowledgements
This work is supported by the National Natural Science Foundation of China (Nos. 11361074,11501141), Foundation of Guizhou Science and Technology Department (Grant No. [2015]2073), Scientific Research Foundation for the introduction of talents of Guizhou Minzu University (No. 15XRY003), and Scientific Research Foundation of Guizhou Minzu University (No. 15XJS009).
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Zhao, J., Sang, C. & Wang, F. New inequalities for the Hadamard product of an M-matrix and an inverse M-matrix. J Inequal Appl 2015, 368 (2015). https://doi.org/10.1186/s13660-015-0893-z
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DOI: https://doi.org/10.1186/s13660-015-0893-z