Trace and Eigenvalue Inequalities for Ordinary and Hadamard Products of Positive Semidefinite Hermitian Matrices

Let A and B be n n positive semidefinite Hermitian matrices, let c and/ be real numbers, let o denote the Hadamard product of matrices, and let Ak denote any k )< k principal submatrix of A. The following trace and eigenvalue inequalities are shown: tr(AoB) <_tr(AoBa), c_<0or_> 1, tr(AoB)a_>tr(AaoBa), 0_a_ 1, A1/a(A o Ba) <_ Al/(Az o B), a <_ /,a O, Al/a[(Aa)k] <_ A1/[(A)k], a <_/,a/ 0. The equalities corresponding to the inequalities above and the known inequalities tr(AS) <_ tr(AaSa), lal >_ 1, and tr(AB) >_ tr(AaSa), I1 _ 1 are thoroughly discussed. Some applications are given. Key words, trace inequality, eigenvalue inequality, Hadamard product, Kronecker product, Schur-convex function, majorization AMS subject classifications. 15A18, 15A39, 15A42, 15A45