The extremal ranks and inertias of matrix expressions with respect to generalized reflexive and anti-reflexive matrices with applications

Abstract

The extremal ranks of matrix expressions of A-BXC and D-EYE*, and the extremal inertias of D-EYE* are discussed, where X and Y are reflexive (or anti-reflexive) and Hermitian reflexive (or anti-reflexive) matrices respectively. For the applications, we derive the extremal ranks of the reflexive and anti-reflexive solutions to AX = B. In addition, we also establish some conditions for the existence of common reflexive and anti-reflexive solutions to AX = B and CXD = E, and conditions for the solvability of some matrix equations and matrix inequalities.