Trace and eigenvalue inequalities for ordinary and Hadamard products of positive semidefinite Hermitian matrices. (English) Zbl 0855.15009
Let \(A\) and \(B\) denote positive semidefinite Hermitian matrices, let \(\alpha\) and \(\beta\) denote real numbers, and let \(\circ\) denote the Hadamard (piecewise) product of matrices. The authors first fully describe the solution of the equation \(\text{tr}(AB)^\alpha=\text{tr}(A^\alpha B^\alpha)\). Then they give inequalities between traces and eigenvalues of \((A\circ B)^\alpha\) and \(A^\alpha\circ B^\alpha\), inequalities between eigenvalues of \(A^\alpha\circ B^\alpha\) and \(A^\beta\circ B^\beta\), and inequalities between eigenvalues of principal submatrices of \(A^\alpha\) and \(A^\beta\).
Reviewer: Z.Dostál (Ostrava)
MSC:
15A42 | Inequalities involving eigenvalues and eigenvectors |
15A18 | Eigenvalues, singular values, and eigenvectors |