Mathematics Faculty Articles
Document Type
Article
Publication Date
1-1-1995
Publication Title
SIAM Journal on Matrix Analysis and Applications
ISSN
0895-4798
Volume
16
Issue/No.
4
First Page
1173
Last Page
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.
NSUWorks Citation
Wang, Bo-Ying and Zhang, Fuzhen, "Trace and Eigenvalue Inequalities of Ordinary and Hadamard Products for Positive Semidefinite Hermitian Matrices" (1995). Mathematics Faculty Articles. 130.
https://nsuworks.nova.edu/math_facarticles/130
DOI
10.1137/S0895479893253616
