Expansion formulas for the inertias of Hermitian matrix polynomials and matrix pencils of orthogonal projectors

This paper gives a group of expansion formulas for the inertias of Hermitian matrix polynomials A−A², I−A² and A−A³ 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.