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

Let Aand Bbe $n \times n$ positive semidefinite Hermitian matrices, let $\alpha $ and $\beta $ be real numbers, let $ \circ $ denote the Hadamard product of matrices, and let $A_k $ denote any $k \times k$ principal submatrix of A. The following trace and eigenvalue inequalities are shown: \[ \operatorname{tr}(A \circ B)^\alpha \leq \operatorname{tr}(A^\alpha \circ B^\alpha ),\quad\alpha \leq 0\,{\text{ or }}\,\alpha \geq 1, \]\[ \operatorname{tr}(A \circ B)^\alpha \geq \operatorname{tr}(A^\alpha \circ B^\alpha ),\quad 0 \leq \alpha \leq 1, \]\[ \lambda^{1/ \alpha } (A^\alpha \circ B^\alpha ) \leq \lambda ^{1/\beta } (A^\beta \circ B^\beta ),\quad\alpha \leq \beta ,\alpha \beta \ne 0, \]\[ \lambda ^{1/\alpha } [(A^\alpha )_k ] \leq \lambda ^{1/\beta } [(A^\beta )_k ],\quad\alpha \leq \beta ,\alpha \beta \ne 0. \]The equalities corresponding to the inequalities above and the known inequalities \[ \operatorname{tr}(AB)^\alpha \leq \operatorname{tr}(A^\alpha B^\alpha ),\quad | \alpha | \geq 1, \] and \[ \operatorname{tr}(AB)^\alpha \geq \operatorname{tr}(A^\alpha B^\alpha ),\quad | \alpha | \leq 1 \] are thoroughly discussed. Some applications are given.