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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. [ABSTRACT FROM AUTHOR]

In this note, for a system of q matrix equations of the form we consider the problem of the existence of Hermitian nonnegative-definite solutions. We offer an alternative with simplification and regularity to the result on necessary and sufficient conditions for the above matrix equations with q = 2 to have a Hermitian nonnegative-definite solution obtained by Zhang [1], who proposed a revised version of Young et al. [2]. Moreover, we give a necessary condition for the general case and then put forward a conjecture, with which at least some special situations are in agreement. [ABSTRACT FROM AUTHOR]