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 ,

for

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.

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.

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

Proof.

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

Proof.

Let

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.

Let

then,

where is an integer.

Proof.

Let

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.

Let

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

Acknowledgment

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