# On Some Matrix Trace Inequalities

- Zübeyde Ulukök
^{1}Email author and - Ramazan Türkmen
^{1}

**2010**:201486

**DOI: **10.1155/2010/201486

© Z. Ulukök and R. Türkmen. 2010

**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 block matrices.

## 1. Introduction and Preliminaries

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 .

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

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

for

where is a positive integer.

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]).

Lemma 2.3 (Cauchy-Schwarz inequality).

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

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

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

where is a positive integer.

## 3. Main Results

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.

Proof.

where .

Theorem 3.2.

Proof.

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.

Proof.

Example 3.4.

## Declarations

### Acknowledgment

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

## Authors’ Affiliations

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## Copyright

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