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 α and β be real numbers, let ∘ 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: \[ tr(A ∘ B)^α ≤ tr(A^α ∘ B^α ),α ≤ 0\,{ or }\,α ≥ 1, \]\[ tr(A ∘ B)^α ≥ tr(A^α ∘ B^α ), 0 ≤ α ≤ 1, \]\[ λ1/ α (A^α ∘ B^α ) ≤ λ 1/β (A^β ∘ B^β ),α ≤ β ,α β ≠ 0, \]\[ λ 1/α [(A^α )_k ] ≤ λ 1/β [(A^β )_k ],α ≤ β ,α β ≠ 0. \] The equalities corresponding to the inequalities above and the known inequalities \[ tr(AB)^α ≤ tr(A^α B^α ), | α | ≥ 1, \] and \[ tr(AB)^α ≥ tr(A^α B^α ), | α | ≤ 1 \] are thoroughly discussed. Some applications are given.