Abstract
Abstract 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.
Highlights
Introduction and preliminariesWe will use the following notation
1, similar results can be obtained by taking similar steps
(A ◦ B)m where m is an positive integer, A ○ B stands for the Hadamard product of A and B
Summary
Where A11 is a square submatrix of A. If A11 is nonsingular, we call A 11 = A22 − A21A−111A12the Schur complement of A11 in A. If A is a positive definite matrix, A11 is nonsingular and A22 ≥ Ã11 ≥ 0. Definition 1.2 (see [2]). A = (aij) Î Cn × n is called a centrosymmetric matrix, if aij = an−i+1,n−j+1,1 ≤ i ≤ n, 1 ≤ j ≤ n, or JnAJn = A, where Jn = (en, en-1,..., e1), ei 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 [2]: Lemma 1.1 (see [2]).
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