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

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.

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