Advances in Linear Algebra & Matrix Theory
Vol.05 No.03(2015), Article ID:59315,7 pages
10.4236/alamt.2015.53009
Matrix Inequalities for the Fan Product and the Hadamard Product of Matrices
Dongjie Gao
Department of Mathematics, Heze University, Heze, China
Email: aizai_2004@126.com
Copyright © 2015 by author and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY).
http://creativecommons.org/licenses/by/4.0/
Received 6 July 2015; accepted 29 August 2015; published 1 September 2015
ABSTRACT
A new inequality on the minimum eigenvalue for the Fan product of nonsingular M-matrices is given. In addition, a new inequality on the spectral radius of the Hadamard product of nonnegative matrices is also obtained. These inequalities can improve considerably some previous results.
Keywords:
M-Matrix, Nonnegative Matrix, Fan Product, Hadamard Product, Spectral Radius, Minimum Eigenvalue
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] .
Cite this paper
DongjieGao, (2015) Matrix Inequalities for the Fan Product and the Hadamard Product of Matrices. Advances in Linear Algebra & Matrix Theory,05,90-97. doi: 10.4236/alamt.2015.53009
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