On Generalizing Trace Minimization Principles, II

Abstract

This paper is concerned with establishing a trace minimization principle for two Hermitian matrix pairs. Specifically, we will answer the question: when is $\inf_X\operatorname{tr}(\widehat AX^{\rm H}AX)$ subject to $\widehat BX^{\rm H}BX=I$ (the identity matrix of apt size) finite? Sufficient and necessary conditions are obtained and, when the infimum is finite, an explicit formula for it is presented in terms of the finite eigenvalues of the matrix pairs. Our results extend Fan's trace minimization principle (1949) for a Hermitian matrix, a minimization principle of Kova\v{c}-Striko and Veseli\'c (1995) for a Hermitian matrix pair, and most recent ones by the authors and their collaborators for a Hermitian matrix pair and a Hermitian matrix.