Physical Sciences and Mathematics Commons^™

Trace And Eigenvalue Inequalities Of Ordinary And Hadamard Products For Positive Semidefinite Hermitian Matrices, Bo-Ying Wang, Fuzhen Zhang

Mathematics Faculty Articles

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 …