Trace inequalities for positive semidefinite matrices with centrosymmetric structure
© Zhao et al; licensee Springer. 2012
Received: 4 July 2011
Accepted: 12 March 2012
Published: 12 March 2012
In this article, we present some results on the Hadamard product of positive semidefinite matrices with centrosymmetric structure. Based on these results, several trace inequalities on positive semidefinite centrosymmetric matrices are obtained.
1 Introduction and preliminaries
We will use the following notation. Let ℂn × nand ℝn × nbe the space of n × n complex and real matrices, respectively. The identity matrix in ℂn × nis denoted by I = I n . Let A T , Ā, A H , and tr(A) denote the transpose, the conjugate, the conjugate transpose, and the trace of a matrix A, respectively. Let Re(a) represent the real part of a. The Frobenius inner product < ·, · > F in ℂm × nover the complex field is defined as follows: < A, B > F = Re(tr(B H A)), for A, B ∈ ℂm × n, i.e., < A, B > is the real part of the trace of B H A. The induced matrix norm is , which is called the Frobenius (Euclidean) norm.
A is further called positive definite, symbolized A > 0, if the strict inequality in (1.1) holds for all non-zero x ∈ ℂ n . An equivalent condition for A ∈ ℂ n to be positive definite is that A is Hermitian and all eigenvalues of A are positive.
Next, we introduce some basic definitions and lemmas.
where A11is a square submatrix of A. If A11is nonsingular, we callthe Schur complement of A11in A.
Note. If A is a positive definite matrix, then A11 is nonsingular and A22 ≥ Ã11 ≥ 0.
where J n = (e n , en-1,..., e1), e i denotes the unit vector with the ith entry 1.
If a matrix is both positive semidefinite and centrosymmetric, we call this matrix positive semidefinite centrosymmetric.
Using the partition of matrix, the central symmetric character of a square centrosymmetric matrix can be described as follows :
Note. In this article, we mainly discuss the case n = 2m. For n is odd, i.e., n = 2m + 1, similar results can be obtained by taking similar steps.
Recently, in , Ulukö k and Tü rkmen proved some matrix trace inequalities for positive semidefinite matrices:
where m is an positive integer, A ○ B stands for the Hadamard product of A and B.
Note. Particularly, if A and B in Lemma 1.2 are both semidefinite matrices, then .
where m is an integer.
2 Main results
Then, A ○ B is a centrosymmetric matrix.
From (2.8) and (2.9), we can see that (2.3) holds. By Lemma 1.1, A ○ B is a centrosymmetric matrix.
Let M = B - J m C and N = B + J m C. Then, M, N are positive semidefinite matrices.
We would like to thank the reviewers for providing valuable comments and suggestions to improve the manuscript. This study was supported by the National Natural Science Foundation of China (No. 60831001).
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