Max-Min Problems on the Ranks and Inertias of the Matrix Expressions A−BXC±(BXC)∗ with Applications

Abstract

We introduce a simultaneous decomposition for a matrix triplet (A,B,C ∗), where A=±A ∗ and (⋅)∗ denotes the conjugate transpose of a matrix, and use the simultaneous decomposition to solve some conjectures on the maximal and minimal values of the ranks of the matrix expressions A−BXC±(BXC)∗ with respect to a variable matrix X. In addition, we give some explicit formulas for the maximal and minimal values of the inertia of the matrix expression A−BXC−(BXC)∗ with respect to X. As applications, we derive the extremal ranks and inertias of the matrix expression D−CXC ∗ subject to Hermitian solutions of a consistent matrix equation AXA ∗=B, as well as the extremal ranks and inertias of the Hermitian Schur complement D−B ∗ A ∼ B with respect to a Hermitian generalized inverse A ∼ of A. Various consequences of these extremal ranks and inertias are also presented in the paper.