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Some inequalities for the Hadamard product of an M-matrix and an inverse M-matrix
Journal of Inequalities and Applications volume 2013, Article number: 16 (2013)
Abstract
Let A and B be nonsingular M-matrices. Some new lower bounds on the minimum eigenvalue for the Hadamard product of A and are given. These bounds improve the corresponding results of Chen (Linear Algebra Appl. 378:159-166, 2004) and Huang (Linear Algebra Appl. 428:1551-1559, 2008) and generalize the corresponding result of Xiang (Linear Algebra Appl. 367:17-27, 2003).
MSC:15A06, 15A15, 15A48.
1 Introduction
For a positive integer n, N denotes the set . The set of all complex matrices is denoted by , and denotes the set of all real matrices throughout.
Let and . We write (>B) if () for all , . If (>0), we say that A is a nonnegative (positive) matrix. The spectral radius of A is denoted by . Let A be an irreducible nonnegative matrix. It is well known that there exists a positive vector u such that , u being called a right Perron eigenvector of A. This guarantees that , where denotes the spectrum of A.
The set is defined by
The simple sign patten of the matrices in has many striking consequences. Let and suppose with and . Then is an eigenvalue of A, every eigenvalue of A lies in the disc , and hence every eigenvalue λ of A satisfies . In particular, a matrix is called an M-matrix if . If , we call A nonsingular M-matrix, and denote the class of nonsingular M-matrices by .
Let , we denote by . The following simple facts are needed for our purpose in proving (see Problems 16 and 19 in Section 2.5 of [1]):
-
(i)
; is called a minimum eigenvalue of A.
-
(ii)
If and is the Perron eigenvalue of the nonnegative matrix , then is a positive real eigenvalue of A.
Let A be an irreducible nonsingular M-matrix. It is well known that there exists a positive vector u such that , u being called a right Perron eigenvector of A.
If , we write , where . Note that for all if . Thus we define the Jacobi iterative matrix of A by . It is easy to check that is nonnegative and (see [2]).
The Hadamard product of and is defined by .
It has been noted [3, 4] that the Hadamard product of an M-matrix B and the inverse of an M-matrix A is again an M-matrix.
In 1991, Horn et al. [[1], p. 375] showed the classical result: if , , , then
Subsequently, Chen [5] improved the bound in (1.1) and obtained the following result:
In 2008, Huang [2] obtained the following result:
This bound in (1.3) improved the bound in (1.1) in some cases. For example, if
then . But in Example 2.1 in this paper.
In practice, the bound of can give a rough estimate before actually solving it and can serve as a check of whether the solution technique for it actually resulted in valid solution. Besides, a good bound of can also help us reduce the computational burden. Therefore, it is necessary to study the bound. In this paper, we present some new lower bounds of the minimum eigenvalue for the Hadamard product of M-matrices, which improve (1.1), (1.2) and (1.3) and generalize the corresponding result of Xiang [6].
2 Main results
In this section, we state and prove our main results. Firstly, we give some lemmas.
Lemma 2.1 (See [[7], Theorem 11])
Let , with . Then, if λ is an eigenvalue of A, there is a pair of positive integers with () such that
Lemma 2.2 (See [[8], Lemma 2.2])
-
(a)
If is an strictly diagonally dominant matrix by row, that is, for any , then exists, and
(b) If is an strictly diagonally dominant matrix by column, that is, for any , then exists, and
Proof We give a simple proof of (a) which is different from that in [8]. Similarly, one can prove (b). Firstly, we prove for all . Suppose not. Let for some j and . We can then assume for all . Since , we have . Thus
which is a contradiction. Hence, holds for all pairs i, j. Thus
that is,
□
Theorem 2.3 Let , and . Then
Proof If both A and B are irreducible. Let and be the right Perron eigenvectors of and A, respectively, i.e., , . Define , where . It is easy to check that C is diagonally dominant by row. It follows from Lemma 2.2, for all , we have
Thus
Let , . Then and
Hence, . Since is an eigenvalue of , we have
Thus, by Lemma 2.1, there exists a pair of positive integers with () such that
From the above inequality and , , we have
Thus, from inequality (2.2), we have
Assume that one of A and B is reducible. It is well known that a matrix in 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 [9]). If we denote by the permutation matrix with , the remaining zero, then both and are irreducible nonsingular M-matrices for any chosen positive real number t sufficiently small such that all the leading principal minors of both and are positive. Now, we substitute and for A and B, respectively, in the previous case, and then letting , the result follows by continuity. □
Using ideas of the proof of Theorem 2.3, we next give a new proof of inequality (2.2) in [5]. Similar to the proof of Theorem 2.3, by the theorem of Gerschgorin, there exist some positive integers such that
From the above inequality and , , we have
Thus, from inequality (2.3), we have
Remark 2.1 We next give a simple comparison between the upper bound in (2.1) and the upper bound in (1.2) and (1.1). Without loss of generality, for , assume that
Thus, we can write (2.4) equivalently as
From (2.1), we have
Thus, from (2.1), (2.5) and the above inequality, we have
Hence, the bound in (2.1) is sharper than the bound in (1.2). According to Remark 2.4 in [5], we know
So, the bound in (2.1) is sharper than the bound in (1.1).
Theorem 2.4 Let , . Then
Proof Suppose that A and B are irreducible, is the diagonal matrix of B and , then is a diagonal matrix with positive diagonal entries, is an irreducible nonnegative matrix and is again an irreducible nonnegative matrix. Since the Jacobi iterative matrix of B is , we have
By the Perron-Frobenius theorem on irreducible nonnegative matrices, there is a positive eigenvector such that . That is,
Thus, we can write (2.8) equivalently as
Set and . It is easy to check that is a strictly diagonally dominant matrix by column. Let . By Lemma 2.2, for all (), we have
Thus
Combining (2.9) with (2.7), we get
Since , we obtain
Thus
Let for positive vectors . Set , then . Hence, . Since is an eigenvalue of , we have
By the theorem of Gerschgorin and (2.10), there exist some positive integers such that
From the above inequality and , , we have
Thus, from inequality (2.11) and (2.12), we have
Assume that one of A and B is reducible. It is well known that a matrix in 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 [9]). If we denote by the permutation matrix with , the remaining zero, then both and are irreducible nonsingular M-matrices for any chosen positive real number t sufficiently small such that all the leading principal minors of both and are positive. Now, we substitute and for A and B, respectively, in the previous case, and then letting , the result follows by continuity. □
Remark 2.2 If is a diagonal matrix, the equality of (2.6) holds. Thus the bound (2.6) is sharp. Since , then
The bound in (2.6) is sharper than the bound in (1.3).
If , according to Theorem 2.4, we can deduce the following corollary.
Corollary 2.5 Let , then
Remark 2.3 Corollary 2.5 is Theorem 2.8 of Xiang [6]. So, Theorem 2.4 generalizes Theorem 2.8 in [6].
If we apply Lemma 2.1 to and , then we have
Since , then
From (2.13) we have the following corollary.
Corollary 2.6 Let , then
Example 2.1 Let A and B be the same as in Example 2.1 in [10]:
It is easy to check that . If we apply Theorem 5.7.31 of [1], we have
If we apply Theorem 9 of [2], we have
If we apply Theorem 2.1 of [10], we have
But if we apply Theorem 2.4, we have
In fact, . Example 2.1 shows that the bound in (2.6) is better than these corresponding bounds in [1, 2, 10].
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Acknowledgements
The work of DZ is supported by the Scholarship Award for Excellent Doctoral Students granted by East China Normal University (No. XRZZ2012021). The work of GC is supported by the National Natural Science Foundation of China (No. 11071079) and the Natural Science Foundation of Zhejiang Province (No. Y6110043). The work of XZ is supported by the National Natural Science Foundation of China (No. 10901056) and the Science and Technology Commission of Shanghai Municipality (No. 11QA1402200).
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Zhou, D., Chen, G., Wu, G. et al. Some inequalities for the Hadamard product of an M-matrix and an inverse M-matrix. J Inequal Appl 2013, 16 (2013). https://doi.org/10.1186/1029-242X-2013-16
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DOI: https://doi.org/10.1186/1029-242X-2013-16