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Abstract: This paper gives a group of expansion formulas for the inertias of Hermitian matrix polynomials , and through some congruence transformations for block matrices, where A is a Hermitian matrix. Then, the paper derives various expansion formulas for the ranks and inertias of some matrix pencils generated from two or three orthogonal projectors and Hermitian unitary matrices. As applications, the paper establishes necessary and sufficient conditions for many matrix equalities to hold, as well as many inequalities in the Löwner partial ordering to hold. [Copyright &y& Elsevier]

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]

Abstract: It is shown that any map which preserves adjacency on hermitian matrices over a finite field is necessary bijective and hence of the standard form. [Copyright &y& Elsevier]