Open Access

On Some Matrix Trace Inequalities

Journal of Inequalities and Applications20102010:201486

DOI: 10.1155/2010/201486

Received: 23 December 2009

Accepted: 14 March 2010

Published: 6 April 2010

Abstract

We first present an inequality for the Frobenius norm of the Hadamard product of two any square matrices and positive semidefinite matrices. Then, we obtain a trace inequality for products of two positive semidefinite block matrices by using https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq1_HTML.gif block matrices.

1. Introduction and Preliminaries

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq2_HTML.gif denote the space of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq3_HTML.gif complex matrices and write https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq4_HTML.gif . The identity matrix in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq5_HTML.gif is denoted https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq6_HTML.gif . As usual, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq7_HTML.gif denotes the conjugate transpose of matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq8_HTML.gif . A matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq9_HTML.gif is Hermitian if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq10_HTML.gif . A Hermitian matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq11_HTML.gif is said to be positive semidefinite or nonnegative definite, written as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq12_HTML.gif , if
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_Equ1_HTML.gif
(1.1)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq13_HTML.gif
is further called positive definite, symbolized https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq14_HTML.gif , if the strict inequality in (1.1) holds for all nonzero https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq15_HTML.gif . An equivalent condition for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq16_HTML.gif to be positive definite is that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq17_HTML.gif is Hermitian and all eigenvalues of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq18_HTML.gif are positive real numbers. Given a positive semidefinite matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq19_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq20_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq21_HTML.gif denotes the unique positive semidefinite https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq22_HTML.gif power of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq23_HTML.gif .
Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq24_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq25_HTML.gif be two Hermitian matrices of the same size. If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq26_HTML.gif is positive semidefinite, we write
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_Equ2_HTML.gif
(1.2)

Denote https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq27_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq28_HTML.gif eigenvalues and singular values of matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq29_HTML.gif , respectively. Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq30_HTML.gif is Hermitian matrix, its eigenvalues are arranged in decreasing order, that is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq31_HTML.gif and if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq32_HTML.gif is any matrix, its singular values are arranged in decreasing order, that is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq33_HTML.gif The trace of a square matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq34_HTML.gif (the sum of its main diagonal entries, or, equivalently, the sum of its eigenvalues) is denoted by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq35_HTML.gif .

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq36_HTML.gif be any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq37_HTML.gif matrix. The Frobenius (Euclidean) norm of matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq38_HTML.gif is
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_Equ3_HTML.gif
(1.3)
It is also equal to the square root of the matrix trace of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq39_HTML.gif that is,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_Equ4_HTML.gif
(1.4)

A norm https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq40_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq41_HTML.gif is called unitarily invariant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq42_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq43_HTML.gif and all unitary https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq44_HTML.gif .

Given two real vectors https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq45_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq46_HTML.gif in decreasing order, we say that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq47_HTML.gif is weakly log majorized by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq48_HTML.gif , denoted https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq49_HTML.gif , if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq50_HTML.gif , and we say that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq51_HTML.gif is weakly majorized by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq52_HTML.gif , denoted https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq53_HTML.gif , if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq54_HTML.gif . We say https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq55_HTML.gif is majorized by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq56_HTML.gif denoted by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq57_HTML.gif , if
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_Equ5_HTML.gif
(1.5)

As is well known, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq58_HTML.gif yields https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq59_HTML.gif (see, e.g., [1, pages 17–19]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq60_HTML.gif be a square complex matrix partitioned as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_Equ6_HTML.gif
(1.6)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq61_HTML.gif is a square submatrix of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq62_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq63_HTML.gif is nonsingular, we call
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_Equ7_HTML.gif
(1.7)
the Schur complement of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq64_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq65_HTML.gif (see, e.g., [2, page 175]). If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq66_HTML.gif is a positive definite matrix, then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq67_HTML.gif is nonsingular and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_Equ8_HTML.gif
(1.8)
Recently, Yang [3] proved two matrix trace inequalities for positive semidefinite matrices https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq68_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq69_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_Equ9_HTML.gif
(1.9)

for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq70_HTML.gif

Also, authors in [4] proved the matrix trace inequality for positive semidefinite matrices https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq71_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq72_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_Equ10_HTML.gif
(1.10)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq73_HTML.gif is a positive integer.

Furthermore, one of the results given in [5] is
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_Equ11_HTML.gif
(1.11)

for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq74_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq75_HTML.gif positive definite matrices, where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq76_HTML.gif is any positive integer.

2. Lemmas

Lemma 2.1 (see, e.g., [6]).

For any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq77_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq78_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq79_HTML.gif .

Lemma 2.2 (see, e.g., [7]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq80_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq81_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq82_HTML.gif then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_Equ12_HTML.gif
(2.1)

Lemma 2.3 (Cauchy-Schwarz inequality).

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq83_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq84_HTML.gif be real numbers. Then,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_Equ13_HTML.gif
(2.2)

Lemma 2.4 (see, e.g., [8, page 269]).

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq85_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq86_HTML.gif are poitive semidefinite matrices, then,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_Equ14_HTML.gif
(2.3)

Lemma 2.5 (see, e.g., [9, page 177]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq87_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq88_HTML.gif are https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq89_HTML.gif matrices. Then,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_Equ15_HTML.gif
(2.4)

Lemma 2.6 (see, e.g., [10]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq90_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq91_HTML.gif are positive semidefinite matrices. Then,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_Equ16_HTML.gif
(2.5)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq92_HTML.gif is a positive integer.

3. Main Results

Horn and Mathias [11] show that for any unitarily invariant norm https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq93_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq94_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_Equ17_HTML.gif
(3.1)
Also, the authors in [12] show that for positive semidefinite matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq95_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq96_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_Equ18_HTML.gif
(3.2)

for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq97_HTML.gif and all unitarily invariant norms https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq98_HTML.gif .

By the following theorem, we present an inequality for Frobenius norm of the power of Hadamard product of two matrices.

Theorem 3.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq99_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq100_HTML.gif be https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq101_HTML.gif -square complex matrices. Then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_Equ19_HTML.gif
(3.3)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq102_HTML.gif is a positive integer. In particular, if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq103_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq104_HTML.gif are positive semidefinite matrices, then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_Equ20_HTML.gif
(3.4)

Proof.

From definition of Frobenius norm, we write
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_Equ21_HTML.gif
(3.5)
Also, for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq105_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq106_HTML.gif , it follows that (see, e.g., [13])
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_Equ22_HTML.gif
(3.6)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_Equ23_HTML.gif
(3.7)
Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq107_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq108_HTML.gif and from inequality (3.7), we write
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_Equ24_HTML.gif
(3.8)
From Lemma 2.1 and Cauchy-Schwarz inequality, we write
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_Equ25_HTML.gif
(3.9)
By combining inequalities (3.7), (3.8), and (3.9), we arrive at
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_Equ26_HTML.gif
(3.10)
Thus, the proof is completed. Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq109_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq110_HTML.gif be positive semidefinite matrices. Then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_Equ27_HTML.gif
(3.11)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq111_HTML.gif .

Theorem 3.2.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq112_HTML.gif be positive semidefinite matrices. For positive real numbers https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq113_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_Equ28_HTML.gif
(3.12)

Proof.

Let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_Equ29_HTML.gif
(3.13)
We know that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq114_HTML.gif , then by using the definition of Frobenius norm, we write
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_Equ30_HTML.gif
(3.14)

Thus, by using Theorem 3.1, the desired is obtained.

Now, we give a trace inequality for positive semidefinite block matrices.

Theorem 3.3.

Let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_Equ31_HTML.gif
(3.15)
then,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_Equ32_HTML.gif
(3.16)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq115_HTML.gif is an integer.

Proof.

Let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_Equ33_HTML.gif
(3.17)
with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq116_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq117_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq118_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq119_HTML.gif (see, e.g., [14]). Let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_Equ34_HTML.gif
(3.18)
with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq120_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq121_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq122_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq123_HTML.gif (see, e.g., [14]). We know that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_Equ35_HTML.gif
(3.)
By using Lemma 2.2, it follows that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_Equ36_HTML.gif
(3.20)
Therefore, we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_Equ37_HTML.gif
(3.21)
As result, we write
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_Equ38_HTML.gif
(3.22)

Example 3.4.

Let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_Equ39_HTML.gif
(3.23)
Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq124_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq125_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq126_HTML.gif From inequality (1.11), for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq127_HTML.gif we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_Equ40_HTML.gif
(3.24)
Also, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq128_HTML.gif , since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq129_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_IEq130_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_Equ41_HTML.gif
(3.25)
Thus, according to this example from (3.24) and (3.25), we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201486/MediaObjects/13660_2009_Article_2083_Equ42_HTML.gif
(3.26)

Declarations

Acknowledgment

This study was supported by the Coordinatorship of Selçuk University's Scientific Research Projects (BAP).

Authors’ Affiliations

(1)
Department of Mathematics, Science Faculty, Selçuk University

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© Z. Ulukök and R. Türkmen. 2010

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