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New inequalities on eigenvalues of the Hadamard product and the Fan product of matrices
Journal of Inequalities and Applications volume 2013, Article number: 433 (2013)
Abstract
In the paper, some new upper bounds for the spectral radius of the Hadamard product of nonnegative matrices, and the low bounds for the minimum eigenvalue of the Fan product of nonsingular M-matrices are given. These new bounds improve existing results, and the estimating formulas are easier to calculate since they only depend on the entries of matrices. Finally, some examples are also given to show that the bounds are better than some previous results.
MSC:65F10, 65F15, 65F50.
1 Introduction
In this paper, for a positive integer n, N denotes the set . () denotes the set of all real (complex) matrices. Let and be two real matrices. We write () if () for all . If (), we say A is a nonnegative (positive) matrix. The spectral radius of A is denoted by . If A is a nonnegative matrix, the Perron-Frobenius theorem guarantees that , where is the set of all eigenvalues of A throughout this paper (see [1]).
For , an matrix A is said to be reducible if there exists a permutation matrix P such that
where B and D are square matrices of order at least one. If no such permutation matrix exists, then A is called irreducible. If A is a complex matrix, then A is irreducible if and only if its single entry is nonzero (see [2]).
According to Ref. [3], a matrix A is called an M-matrix if there exists an nonnegative real matrix P and a nonnegative real number α such that and , where I is the identity matrix. Moreover, if , A is called a nonsingular M-matrix; if , we call A a singular M-matrix.
In addition, a matrix is called Z-matrix if all of it off-diagonal entries are negative and denoted by . For convenience, the following simple facts are needed (see Problems 16, 19 and 28 in Section 2.5 of [3]), where , and is denoted by the set of all nonsingular M-matrices (see [1]):
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1.
;
-
2.
If and , then ;
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3.
If , then is the Perron eigenvalue of the nonnegative matrix , and is a positive real eigenvalue of A.
Let A be an irreducible nonnegative matrix. It is well known that there exist positive vectors u and v such that and , where u and v are right and left Perron eigenvectors of A, respectively.
The Hadamard product of and is defined by .
For two real matrices , the Fan product of A and B is denoted by and is defined by
Obviously, if , then is also an M-matrix (see [2]).
We define
throughout the paper.
The paper is organized as follows. Firstly, for two nonnegative matrices A and B, we exhibit some new upper bounds for in Section 2. In Section 3, some new lower bounds for of M-matrices are presented. Finally, some examples are given to illustrate our results.
2 Some upper bounds for the spectral radius of the Hadamard product of two nonnegative matrices
Firstly, in ([3], p.358), there is a simple estimate for : if , , and , then
Recently, Fang [4] gave an upper bound for , that is,
which is smaller than the bound in ([3], p.358).
Liu and Chen [2] improved (2.2) and gave the following result:
Recently, some elaborate new bounds were also presented in [5], which in some cases give better estimates for the spectral radius of the Hadamard product of two nonnegative matrices.
In this section, based on the idea of [5], we present some new upper bounds on for nonnegative matrices A and B which improve the above results. The new estimating formulae also only depend on the entries of matrices A and B.
Lemma 2.1 [6]
Let be an arbitrary complex matrix, and let be positive real numbers, then all the eigenvalues of A lie in the region
Lemma 2.2 [7]
Let , and let be positive real numbers, then all the eigenvalues of A lie in the region
Next, we present a new estimating formula of the upper bounds of which is easier to calculate.
Theorem 2.1 If and are nonnegative matrices, then
Proof It is evident that inequality (2.6) holds with equality for . Therefore, we assume that and give two cases to prove this problem.
Case 1. Suppose that is irreducible. Obviously A and B are also irreducible. By Lemma 2.1, there exists () such that
i.e.,
Thus, we have that
So, conclusion (2.6) holds.
Case 2. If is reducible. We may denote by the permutation matrix with
the remaining being zero, then both and are nonnegative irreducible matrices for any sufficiently small positive real number ε. Now we substitute and for A and B, respectively, in the previous Case 1, and then letting , the result (2.6) follows by continuity. □
Theorem 2.2 If and are nonnegative matrices, then
Proof Similarly, inequality (2.7) holds with equality for . Therefore, we assume that and give two cases to prove this problem.
Case 1. Suppose that is irreducible. Obviously, A and B are also irreducible. By Lemma 2.2, there exists a pair of positive integers with () such that
From inequality (2.8) and (see [8]), for any , we have
Thus, by solving quadratic inequality (2.9), we have that
i.e., conclusion (2.7) holds.
Case 2. If is reducible. We may denote by the permutation matrix with
the remaining being zero, then both and are nonnegative irreducible matrices for any sufficiently small positive real number ε. Now we substitute and for A and B, respectively, in the previous Case 1, and then letting , the result (2.7) follows by continuity. □
Remark 2.1 Next, we give a comparison between inequality (2.6) and inequality (2.7). Without loss of generality, for , we assume that
Thus, we can rewrite (2.10) as
From (2.11), we have that
Thus, from (2.7) and the above inequality, we can obtain
Hence, the bound in (2.7) is sharper than the bound in (2.6).
Example 2.1 [1]
Let
If we apply (2.1), we have
If we apply (2.2), we have
If we apply (2.3), we have
If we apply Theorem 2.1, we get
If we apply Theorem 2.2, we obtain that
In fact, . The example shows that the bounds in Theorem 2.1 and Theorem 2.2 are better than the existing bounds.
3 Inequalities for the Fan product of two M-matrices
Firstly, let us recall some results. It is known (p.359, [3]) that the following classical result is given. If are M-matrices, then
In 2007, Fang improved (3.1) in Remark 3 of Ref. [4] and gave a new lower bound for , that is,
Subsequently, Liu and Chen [2] gave a sharper bound than (3.2), i.e.,
Next, we exhibit a new lower bound on the minimum eigenvalue of the Fan product of nonsingular M-matrices.
Theorem 3.1 If and are nonsingular M-matrices, then
Proof It is evident that inequality (3.4) holds with equality for . Therefore, we assume that and give two cases to prove this problem.
Case 1. Suppose that is irreducible. Obviously, A and B are also irreducible. By Lemma 2.1, there exists i () such that
From inequality (3.5) and (see [8]), for any , we have
Thus, we can obtain that
i.e., the conclusion (3.4) holds.
Case 2. If 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 [8]). If we denote by the permutation matrix with
the remaining being zero, then both and are irreducible nonsingular M-matrices for any sufficiently small positive real number ε 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 1, and then letting , the result (3.4) follows by continuity. □
Theorem 3.2 If and are nonsingular M-matrices, then
Proof Obviously, inequality (3.7) holds with equality for . Therefore, we assume that and give two cases to prove this problem.
Case 1. Suppose that is irreducible, then A and B are also irreducible. By Lemma 2.2, there exists a pair of positive integers with () such that
From inequality (3.8) and (see [8]), for any , we have
Thus, by solving quadratic inequality (3.9), we have that
i.e., conclusion (3.7) holds.
Case 2. Similarly, if 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 [8]). If we denote by the permutation matrix with
the remaining being zero, then both and are irreducible nonsingular M-matrices for any sufficiently small positive real number ε 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 1, and then letting , the result (3.7) follows by continuity. □
Remark 3.1 Similarly, by solving quadratic inequality (3.9) and the same proof as Theorem 3.2, one can also obtain an upper bound on the :
Remark 3.2 Next, we give a comparison between inequality (3.4) and inequality (3.7). Without loss of generality, for , we assume that
Thus, we can rewrite (3.10) as
From (3.11), we have that
Thus, from (3.7) and the above inequality, we can obtain
Hence, the bound in (3.7) is sharper than the bound in (3.4).
Next, let us consider a simple example.
Example 3.1 [1]
Consider two M-matrices
By calculation, we obtain that . If we apply (3.1), we can get that
If we apply (3.2), we have that
If we apply (3.3), we have
If we apply (3.4), we have that
If we apply (3.7), we get that
From the above example, inequality (3.7) is obviously the best one corresponding to inequalities (3.1), (3.2), (3.3) and (3.4).
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Acknowledgements
This paper is supported by the National Natural Science Foundation of China (11026085, 11101071, 11271001, 51175443) and the Fundamental Research Funds for China Scholarship Council.
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Guo, QP., Li, HB. & Song, MY. New inequalities on eigenvalues of the Hadamard product and the Fan product of matrices. J Inequal Appl 2013, 433 (2013). https://doi.org/10.1186/1029-242X-2013-433
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DOI: https://doi.org/10.1186/1029-242X-2013-433