## Mathematics Faculty Articles

Article

1-1-1995

#### Publication Title

SIAM Journal on Matrix Analysis and Applications

0895-4798

16

4

1173

1183

#### Abstract

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.

#### DOI

10.1137/S0895479893253616

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