A survey on rank and inertia optimization problems of the matrix-valued function $A + BXB^{*}$

Abstract

This paper is concerned with some rank and inertia optimization problems of the Hermitian matrix-valued functions A + BXB * subject to restrictions. We first establish several groups of explicit formula for calculating the maximum and minimum ranks and inertias of matrix sum A + X subject to a Hermitian matrix X that satisfies a fixed-rank and semi-definiteness restrictions by using some discrete and matrix decomposition methods. We then derive formulas for calculating the maximum and minimum ranks and inertias of the matrix-valued function A + BXB * subject to a Hermitian matrix X that satisfies a fixed-rank and semi-definiteness restrictions, and show various properties A + BXB * from these ranks and inertias formulas. In particular, we give necessary and sufficient conditions for the equality A + BXB * = 0 and the inequality A + BXB * 0 (0, ≺ 0, 0) to hold respectively for these specified Hermitian matrices X.