Trace inequalities for positive semidefinite matrices with centrosymmetric structure
Journal of Inequalities and Applications volume 2012, Article number: 62 (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 AT, Ā, AH, 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(BHA)), for A, B ∈ ℂm × n, i.e., < A, B > is the real part of the trace of BHA. The induced matrix norm is , which is called the Frobenius (Euclidean) norm.
A matrix A ∈ ℂn × nis Hermitian if AH= A, and A is called positive semidefinite, written as A ≥ 0 (see [1, p. 159]), if
A is further called positive definite, symbolized A > 0, if the strict inequality in (1.1) holds for all non-zero x ∈ ℂn. An equivalent condition for A ∈ ℂnto be positive definite is that A is Hermitian and all eigenvalues of A are positive.
Let A and B be two Hermitian matrices of the same size. If A - B is positive semidefinite, we write
Next, we introduce some basic definitions and lemmas.
Definition 1.1 (see ). Let A be a square complex matrix partitioned as
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.
Definition 1.2 (see ). A = (a ij ) ∈ ℂn × nis called a centrosymmetric matrix, if
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 :
Lemma 1.1 (see ). Let A = (a ij ) ∈ ℂn × n(n = 2m) be centrosymmetric. Then, A has the following form,
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:
Lemma 1.2 (see ). Let A, B ∈ ℂn × n. Then,
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 .
Lemma 1.3 (see ). Let A i ∈ ℂn × n, (i = 1,2,...,k) be semidefinite matrices. Then, for positive real numbers s, m, t
Lemma 1.4 (see ). Let. Then,
where m is an integer.
2 Main results
Lemma 2.1. Let A = (a ij )n × n, B = (b ij )n × n(n = 2m) be two centrosymmetric matrices with the following form:
Then, A ○ B is a centrosymmetric matrix.
Proof. By the definition of Hadamard product,
We shall prove the following
we have the following
From (2.5) and (2.7), it is clear that
Similarly, we can prove that
From (2.8) and (2.9), we can see that (2.3) holds. By Lemma 1.1, A ○ B is a centrosymmetric matrix.
Lemma 2.2 (see ) Let A = (a ij )n × n(n = 2m) a positive semidefinite centrosymmetric matrix with the following form
Let M = B - J m C and N = B + J m C. Then, M, N are positive semidefinite matrices.
Theorem 2.1 Let A, B ∈ ℂn × n(n = 2m) be two positive semidefinite centrosymmetric matrices with the same form as in (2.1). Let
Then, the following inequality holds:
Proof. Since A, B are centrosymmetric matrices, by Lemma 2.1, A ○ B is centrosymmetric and
From Lemma 1.1,
P is orthogonal, then
Similarly from Lemma 1.1,
Since both A and B are positive semidefinite matrices, by Lemma 1.2
Theorem 2.2. Let A ∈ ℂn × n(n = 2m) be postive semidefinite centroysymmetric with the form:
Let M = B - J m C, N = B + J m C. Then, for positive integers s, m, t, the following two equalities hold
Proof. By Lemma 1.1,
Since A is positive semidefinite, we have
From Lemma 2.2, M and N are positive semidefinite. Combining Lemma 1.3, we have
Theorem 2.3 Let A, B ∈ ℂn × n(n = 2m) are positive semidefinite centrosymmetric matrices with the same form as in (2.1). Let
Then, the following inequality holds
Proof. From Lemma 1.1, there exists an orthogonal matrix P such that
According to Definition 1.1,
From Lemma 1.4,
Since P is orthogonal, we have tr(PTABP)m= tr(AB)m. By Lemma 1.4, the following holds
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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).
The authors declare that they have no competing interests.
DZ carried out studies on the linear algebra and matrix theory with applications, and drafted the manuscript. HL helped prove some lemmas and theorems. ZG read the manuscript carefully and gave valuable suggestions and comments. All authors read and approved the final manuscript.
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Zhao, D., Li, H. & Gong, Z. Trace inequalities for positive semidefinite matrices with centrosymmetric structure. J Inequal Appl 2012, 62 (2012). https://doi.org/10.1186/1029-242X-2012-62