Matrix Inequalities for the Fan Product and the Hadamard Product of Matrices ()
1. Introduction
Let, and. We write if for any. If, A is called a nonnegative matrix, and if A > 0, A is called a positive matrix. The spectral radius of a nonnegative matrix A is denoted by.
We denote by Zn the class of all n × n real matrices, all of whose off-diagonal entries are nonpositive. A matrix is called an M-matrix if there exists a nonnegative matrix B and a nonnegative real number s, such that with, where I is the identity matrix. If (resp.,), then the M-matrix A is nonsingular (resp., singular) (see [1] [2] ). Denote by Mn the set of nonsingular M-matrices. We define, where denotes the spectrum of A.
The Fan product of two matrices and is the matrix, where
If, then so is. In ([2] , p. 359), a lower bound for was given: if, then.
If, and, we write, where. Thus we define. Obviously, JA is nonnegative. Recently, some authors gave some lower bounds of (see [3] -[8] ). In [4] , Huang obtained the following result for,
(1)
The bound of (1) is better than the bound in ([2] , p. 359).
In [7] , Liu gave a lower bound of,
(2)
where. The bound of (2) is better than the one of (1).
For a nonnegative matrix, let, where. We denote, where,
The Hadamard product of two matrices and is the matrix . For two nonnegative matrices A and B, recently, some authors gave several new upper bounds of (see [3] -[7] [9] ). In [4] , Huang obtained the following result for,
1) If, then
(3)
2) If or for some i0, but, then
(4)
3) If and, then
(5)
4) If and for some i0, j0, then the upper bound of is the maximum value of the upper bounds of the inequalities in (3)-(5).
The bound of in [4] is better than that in ([2] , p. 358).
In [7] , Liu gave a new upper bound of,
1) If, then
(6)
where.
2) If and or and for some, but, then
(7)
3) If and, then
(8)
4) If and for some i0, j0, then the upper bound of is the maximum value of the upper bounds of the inequalities in (6)-(8).
The bound of in [7] is better than that in [4] .
The paper is organized as follows. In Section 2, we give a new lower bound of. In Section 3, we present a new upper bound of.
2. Inequalities for the Fan Product of Two M-Matrices
In this section, we will give a new lower bound of.
If and, we write for the k-th Hadamard power of A. If and, we write.
Lemma 1. [7] Let, and let be two positive diagonal matrices. Then
Lemma 2. [2] If is a nonnegative matrix and, then
Theorem 1. Let and. Then
where.
It is evident that the Theorem holds with equality for n = 1. Next, we assume that.
(1) First, we assume that is irreducible matrix, then A and B are irreducible. Obviously JA and JB are also irreducible and nonnegative, so and are nonnegative irreducible matrices. Then there exist two
positive vectors and such that and. Let
Then we have and, that is
Let and in which U and V are the nonsingular diagonal matrices
and. Then, we have
It is easy to see that, , and VU are nonsingular since V and U are. From Lemma 1, we have
Thus, we obtain, and
We next consider the minimum eigenvalue of. Let. Then we have that. By Theorem 1.23 of [10] , there exist, , such that
By Hölder’s inequality, we have
Then, we have
Since, then
Hence,
i.e.,
(2) Now, assume that is reducible. It is well known that a matrix in Zn is a nonsingular M-matrix if and only if all its leading principal minors are positive (see [11] ). If we denote by the n × n permutation matrix with, the remaining tij zero, then both and are irreducible nonsingular M-matrix for any chosen positive real number, 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 1. By Lemma 2, the bound in Theorem 1 is better than that in Theorem 4 of [8] and Theorem 2 of [7] .
Example 1. Let
By calculating with Matlab 7.1, it is easy to show that.
Applying Theorem 4 of [4] , Theorem 3.1 of [5] , Theorem 2 of [7] , and Theorem 3.1 of [8] , we have, , , and, respectively. But, if we apply Theorem 1, we have
The numerical example shows that the bound in Theorem 1 is better than that in Theorem 4 of [4] , Theorem 3.1 of [5] , Theorem 2 of [7] , and Theorem 3.1 of [8] .
3. Inequalities for the Hadamard Product of Two Nonnegative Matrices
In this section, we will give a new upper bound of for nonnegative matrices A and B. Similar to [7] , for, write Q = A − D, where. We denote with, where
Note that is nonnegative, and if,. For, let, where
Similarly, the nonnegative matrix is defined.
Lemma 3. [2] Let, and let be diagonal matrices. Then
Lemma 4. [12] Let be a nonnegative matrix. Then
Theorem 2. Let, and. Then
1) If, then
(9)
where.
2) If and or and for some, but, then
(10)
3) If and, then
(11)
4) If and for some i0, j0, then the upper bound of is the maximum value of the upper bounds of the inequalities in (9)-(11).
Proof. It is evident that 4) holds with equality for n = 1. Next, we assume that.
(1) First, we assume that is irreducible matrix, then A and B are irreducible. Obviously and are also irreducible and nonnegative, so and are nonnegative irreducible matrices. Then there exist two positive vectors and such that and. Let
Then we have and, that is
Let and in which U and V are the nonsingular diagonal matrices
and. Then we have
It is easy to see that, , and VU are nonsingular since V and U are. From Lemma 4, we have
Thus, we obtain, and
We next consider the minimum eigenvalue of. For nonnegative irreducible matrices
and, by definition of the Hadamard product of and, Hölder’s inequality, and Lemma 5, we have
Thus, we obtain
1) If, then
2) If and or and for some i0, j0, but, then
3) If and, then
4) If and for some i0, j0, then the upper bound of is the maximum value of the upper bounds of the inequalities in (9)-(11).
(2) Now, we assume that is reducible. If we denote by the n × n permutation matrix with, the remaining tij = 0, then both and are irreducible nonsingular matrices for any chosen positive real number. Now, we substitute and for A and B, respectively, in the previous case, and then letting, the result follows by continuity.
Remark 2. By Lemma 2, the bound in Theorem 2 is better than that in Theorem 6 of [6] and Theorem 3 of [9] .
Example 2. Let
By calculation with Matlab 7.1, we have, , , , and.
If we apply Theorem 6 of [4] , Theorem 3 of [7] , and Theorem 2.2 of [9] , we have, , and, respectively. But, if we apply Theorem 2, we have
The numerical example shows that the bound in Theorem 2 is better than that in Theorem 6 of [4] , Theorem 3 of [7] , and Theorem 2.2 of [9] .