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Open Access Research Article

On Some Matrix Trace Inequalities

Zübeyde Ulukök* and Ramazan Türkmen

Author Affiliations

Department of Mathematics, Science Faculty, Selçuk University, 42003 Konya, Turkey

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Journal of Inequalities and Applications 2010, 2010:201486  doi:10.1155/2010/201486

The electronic version of this article is the complete one and can be found online at: http://www.journalofinequalitiesandapplications.com/content/2010/1/201486

Received:23 December 2009
Revisions received:4 March 2010
Accepted:14 March 2010
Published:6 April 2010

© 2010 The Author(s).

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 block matrices.

1. Introduction and Preliminaries

Let denote the space of complex matrices and write . The identity matrix in is denoted . As usual, denotes the conjugate transpose of matrix . A matrix is Hermitian if . A Hermitian matrix is said to be positive semidefinite or nonnegative definite, written as , if


is further called positive definite, symbolized , if the strict inequality in (1.1) holds for all nonzero . An equivalent condition for to be positive definite is that is Hermitian and all eigenvalues of are positive real numbers. Given a positive semidefinite matrix and , denotes the unique positive semidefinite power of .

Let and be two Hermitian matrices of the same size. If is positive semidefinite, we write


Denote and eigenvalues and singular values of matrix , respectively. Since is Hermitian matrix, its eigenvalues are arranged in decreasing order, that is, and if is any matrix, its singular values are arranged in decreasing order, that is, The trace of a square matrix (the sum of its main diagonal entries, or, equivalently, the sum of its eigenvalues) is denoted by .

Let be any matrix. The Frobenius (Euclidean) norm of matrix is


It is also equal to the square root of the matrix trace of that is,


A norm on is called unitarily invariant for all and all unitary .

Given two real vectors and in decreasing order, we say that is weakly log majorized by , denoted , if , and we say that is weakly majorized by , denoted , if . We say is majorized by denoted by , if


As is well known, yields (see, e.g., [1, pages 17–19]).

Let be a square complex matrix partitioned as


where is a square submatrix of . If is nonsingular, we call


the Schur complement of in (see, e.g., [2, page 175]). If is a positive definite matrix, then is nonsingular and


Recently, Yang [3] proved two matrix trace inequalities for positive semidefinite matrices and ,



Also, authors in [4] proved the matrix trace inequality for positive semidefinite matrices and ,


where is a positive integer.

Furthermore, one of the results given in [5] is


for and positive definite matrices, where is any positive integer.

2. Lemmas

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

For any and .

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

Let then


Lemma 2.3 (Cauchy-Schwarz inequality).

Let and be real numbers. Then,


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

If and are poitive semidefinite matrices, then,


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

Let and are matrices. Then,


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

Let and are positive semidefinite matrices. Then,


where is a positive integer.

3. Main Results

Horn and Mathias [11] show that for any unitarily invariant norm on


Also, the authors in [12] show that for positive semidefinite matrix , where


for all and all unitarily invariant norms .

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

Theorem 3.1.

Let and be -square complex matrices. Then


where is a positive integer. In particular, if and are positive semidefinite matrices, then



From definition of Frobenius norm, we write


Also, for any and , it follows that (see, e.g., [13])



Since for and from inequality (3.7), we write


From Lemma 2.1 and Cauchy-Schwarz inequality, we write


By combining inequalities (3.7), (3.8), and (3.9), we arrive at


Thus, the proof is completed. Let and be positive semidefinite matrices. Then


where .

Theorem 3.2.

Let be positive semidefinite matrices. For positive real numbers





We know that , then by using the definition of Frobenius norm, we write


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

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

Theorem 3.3.





where is an integer.




with . Then (see, e.g., [14]). Let


with , , . Then (see, e.g., [14]). We know that


By using Lemma 2.2, it follows that


Therefore, we get


As result, we write


Example 3.4.



Then From inequality (1.11), for we get


Also, for , since and , we get


Thus, according to this example from (3.24) and (3.25), we get



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


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