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

We introduce a simultaneous decomposition for a matrix triplet (A,B,C (au)), where A=+/- A (au) and (a <...)(au) 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)(au) 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)(au) with respect to X. As applications, we derive the extremal ranks and inertias of the matrix expression D-CXC (au) subject to Hermitian solutions of a consistent matrix equation AXA (au)=B, as well as the extremal ranks and inertias of the Hermitian Schur complement D-B (au) A (similar to) B with respect to a Hermitian generalized inverse A similar to of A. Various consequences of these extremal ranks and inertias are also presented in the paper.