TRACE AND EIGENVALUE INEQUALITIES FOR ORDINARY AND HADAMARD-PRODUCTS OF POSITIVE SEMIDEFINITE HERMITIAN MATRICES

Let A and B be n x n positive semidefinite Hermitian matrices, let alpha and beta be real numbers, let o denote the Hadamard product of matrices, and let A(k) denote any k x k principal submatrix of A. The following trace and eie;envalue inequalities are shown: tr(A o B)(alpha) less than or equal to tr(A(alpha) o B-alpha), alpha less than or equal to 0 or alpha greater than or equal to 1, tr(A o B)(alpha) greater than or equal to tr(A(alpha) o B-alpha), 0 less than or equal to alpha less than or equal to 1, lambda(1/alpha) (A(alpha) o B-alpha) less than or equal to lambda(1/beta) (A(beta) o B-beta), alpha less than or equal to beta, alpha beta not equal 0, lambda(1/alpha) [(A(alpha))(k)] less than or equal to lambda(1/beta) [(A(beta))(k)], alpha less than or equal to beta, alpha beta not equal 0. The equalities corresponding to the inequalities above and the known inequalities tr(AB)(alpha) less than or equal to tr(A(alpha) o B-alpha), /alpha/ greater than or equal to 1, and tr(AB)(alpha) greater than or equal to tr(A(alpha)B(alpha)), /alpha/ less than or equal to 1 are thoroughly discussed. Some applications are given.