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Two inequalities for the Hadamard product of matrices
Journal of Inequalities and Applications volume 2012, Article number: 122 (2012)
Abstract
Using a estimate on the Perron root of the nonnegative matrix in terms of paths in the associated directed graph, two new upper bounds for the Hadamard product of matrices are proposed. These bounds improve some existing results and this is shown by numerical examples.
MSC 2010: 15A42; 15B34
1 Introduction
Let M n denote the set of all n × n complex matrices and N denote the set {1, 2, ..., n}. Let A = (a ij ), B = (b ij ) ∈ M n . If a ij - b ij ≥ 0, we say that A ≥ B, and if a ij ≥ 0, we say that A is nonnegative. The spectral radius of A is denoted by ρ(A). If A is a nonnegative matrix, the Perron-Frobenius theorem guarantees that ρ(A) ∈ σ(A), where σ(A) denotes the spectrum of A.
If there does not exist a permutation matrix P such that
where A1, A2 are square matrices, then A is called irreducible. Let A be an irreducible nonnegative matrix. It is well known that there exists a positive vector u such that Au = ρ(A)u. The Hadamard product of A, B is defined as A ○ B = (a ij b ij ) ∈ M n . Let A ∈ M n , and let
denote the absolute row sums and the deleted absolute row sums of A, respectively.
Let ς(A) represent the set of all simple circuits in the digraph Γ(A) of A. Recall that a circuit of length k in Γ(A) is an ordered sequence γ = (i1, ..., i k , ik+1), where i1, ..., i k ∈ N are all distinct, ik+1= i1. The set {i1, ..., i k } is called the support of γ and is denoted by . The length of the circuit is denoted by |γ|.
In [1], there is a simple estimate for ρ(A ○ B): if A ≥ 0, B ≥ 0, then ρ(A ○ B) ≤ ρ(A)ρ(B).
Recently, using the Gersgorin theorem that involves only elements in one row or column of the matrix, Fang [2] and Huang [3] gave new estimates for ρ(A ○ B) that were better than the result of [1]. Using the Brauer theorem that involves elements in two rows of the matrix at a time, the authors of [4, 5] derived new upper bounds for ρ(A ○ B) that improved the results of [2, 3]. As we all know, besides Gersgorin theorem and Brauer theorem, Brualdi theorem is also an important eigenvalue inclusion theorem and it involves more elements of the matrix than the other two theorems. In view of this, Liu [4] proposed the following problem: Could we get some new estimate better than the previous results using Brualdi theorem? In this paper, we give affirmative conclusions. Two new upper bounds for ρ(A ○ B) are provided. These bounds improve some existing results and numerical examples illustrate that our results are superior.
2 Main results
First, we give some lemmas which are useful for obtaining the main results.
Lemma 2.1 [6] Let A ∈ M n be a nonnegative matrix. If A k is a principal submatrix of A, then ρ(A k ) ≤ ρ(A). If A is irreducible and A k ≠ A, then ρ(A k ) < ρ(A).
Lemma 2.2 [7] Let A ∈ M n be a nonnegative matrix, and let ς(A) ≠ ∅. Then for any diagonal matrix D with positive diagonal entries, we have
Lemma 2.3 [4] Let A, B ∈ M n . If E; F are diagonal matrices of order n, then
Theorem 2.1 Let A, B ∈ M n , and A ≥ 0, B ≥ 0. Then
Proof. If A ○ B is irreducible, then A and B are irreducible. From Lemma 2.1, we have
Since A = (a ij ), B = (b ij ) are nonnegative irreducible, there exist two positive vectors u, v such that Au = ρ(A)u, Bv = ρ(B)v. Thus, we have
and
Define U = diag(u1, ..., u n ), V = diag(v1, ..., v n ). Let , . From (2) and (3), we have
and
Let D = VU. According to Lemma 2.2, for the positive diagonal matrix D, we have
Using Lemma 2.3, we have
Then,
So, we have
If A ○ B is reducible, then one of A and B is reducible. If we denote by P = (p ij ) the n × n permutation matrix with p12 = p23 = · · · = pn 1= 1, the remaining p ij = 0, then both A + tP and B + tP are nonnegative irreducible matrices for any chosen positive real numbers t. Now, we substitute A + tP and B+tP for A and B, respectively in the previous case, and then letting t → 0, the result follows by continuity.
Two bounds for ρ(A ○ B) given in [2] and [4], respectively, are
and
Next, we give a simple comparison between (1) and (4). It is easy to see
Then the bound (1) is better than the bound (4). From the difference between (1) and (5), we could not verify that (1) is better than (5) in theoretical analysis, but the following numerical example shows that the result derived in Theorem 2.1 is better than (4) and (5).
Example 2.1. Consider two 4 × 4 nonnegative matrices
It is easy to calculate that ρ(A ○ B) = ρ(A) = 5.4983. By inequalities (4) and (5), we have
and ρ(A ○ B) = 11.6478, and by Theorem 2.1, we get
Next, we will give the second inequality for ρ(A ○ B). For A ≥ 0, write L = A - D, where D = diag(a11, ..., a nn ). with We denote with D1 = diag(d ii ), where
Then, J A is nonnegative, and J A = A if a ii = 0 for all i. For B ≥ 0, let D2 = diag(s ii ), with
Then the nonnegative matrix J B can be similarly defined.
Theorem 2.2 Let A, B ∈ M n , and A ≥ 0, B ≥ 0. Then
Proof. If A ○ B is nonnegative irreducible, then A and B are irreducible. Since J A and J B are also nonnegative irreducible, there exist two positive vectors x, y such that J A x = ρ(J A )x, J B y = ρ(J B )y. So, we have
Let , and in which and are nonsingular diagonal matrices and .
From Lemma 2.3, we have
and then
Let . Then for the positive diagonal matrix W, it follows from Lemma 2.2 that
If A ○ B is reducible, then substituting A + tP and B + tP for A and B, respectively in the previous case, letting t → 0, the result is derived.
The bounds for ρ(A ○ B) obtained in [3] and [5], respectively, are
and
It can be easily verified that the bound (6) is better than the bound (7). Here too, we could not give the comparison between (6) and (8), but the following example shows that the result obtained in Theorem 2.2 is better than (7) and (8).
Example 2.2. Let
Then
It is clear that ρ(J A ) = 0.8182, ρ(J B ) = 1.1258, and ρ(A ○ B) = 6.3365. By (7) and (8), we have
and ρ(A ○ B) = 9.6221, and by Theorem 2.2, we get
References
Horn RA, Johnson CR: Topics in Matrix Analysis. Cambridge University Press, Cambridge; 1985.
Fang MZ: Bounds on eigenvalues of the Hadamard product and the Fan product of matrices. Linear Algebr Appl 2007, 425: 7–15. 10.1016/j.laa.2007.03.024
Huang R: Some inequalities for the Hadamard product and the Fan product of matrices. Linear Algebr Appl 2008, 428: 1551–1559. 10.1016/j.laa.2007.10.001
Liu QB, Chen GL: On two inequalities for the Hadamard product and the Fan product of matrices. Linear Algebr Appl 2009, 431: 974–984. 10.1016/j.laa.2009.03.049
Liu QB, Chen GL, Zhao LL: Some new bounds on the spectral radius of matrices. Linear Algebr Appl 2010, 432: 936–948. 10.1016/j.laa.2009.10.006
Berman A, Plemmons RJ: Nonnegative Matrices in the Mathematical Sciences. SIAM, Philadelphia; 1994.
Kolotilina LY: Bounds for the Perron root, singularity/nonsingularity conditions, and Eigenvalue inclusion sets. Numer Algorithm 2006, 42: 247–280. 10.1007/s11075-006-9041-7
Acknowledgements
The author wishes to thank Prof. Guoliang Chen and Dr. Qingbin Liu for their help. This research is financed by NSFC(10971070,11071079).
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Zhao, L. Two inequalities for the Hadamard product of matrices. J Inequal Appl 2012, 122 (2012). https://doi.org/10.1186/1029-242X-2012-122
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DOI: https://doi.org/10.1186/1029-242X-2012-122