Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article On Some Matrix Trace Inequalities ¨ ¨ ¨ Zubeyde Ulukok and Ramazan Turkmen | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010 Article ID 201486 8 pages doi 2010 201486 Research Article On Some Matrix Trace Inequalities Ziibeyde Ulukok and Ramazan Turkmen Department of Mathematics Science Faculty Selcuk University 42003 Konya Turkey Correspondence should be addressed to Zubeyde Ulukok zulukok@ Received 23 December 2009 Revised 4 March 2010 Accepted 14 March 2010 Academic Editor Martin Bohner Copyright 2010 Z. Ulukok and R. Turkmen. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 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 2 X 2 block matrices. 1. Introduction and Preliminaries Let Mm n denote the space of m X n complex matrices and write Mn Mn n. The identity matrix in Mn is denoted In. As usual A A denotes the conjugate transpose of matrix A. A matrix A e Mn is Hermitian if A A. A Hermitian matrix A is said to be positive semidefinite or nonnegative definite written as A 0 if x Ax 0 Vx e Cn. A is further called positive definite symbolized A 0 if the strict inequality in holds for all nonzero x e Cn. An equivalent condition for A e Mn to be positive definite is that A is Hermitian and all eigenvalues of A are positive real numbers. Given a positive semidefinite matrix A and p 0 Ap denotes the unique positive semidefinite pth power of A. Let A and B be two Hermitian matrices of the same size. If A - B is positive semidefinite we write A B or B A. Denote A1 A . . Xn A and s1 A . . sn A eigenvalues and singular values of matrix A respectively. Since A is Hermitian matrix its eigenvalues are arranged in decreasing order that
