Expansion formulas for the inertias of Hermitian matrix polynomials and matrix pencils of orthogonal projectors (Q624568)

For a Hermitian matrix \(A, \) some formulas for the partial inertias of the matrix polynomials \(A-A^2, I-A^2\) and \(A-A^3\) in terms of the partial inertias of \(A, I+A\) and \(I-A\) are proved. The author uses congruence transformations for block matrices to derive these results. Expansion formulas for rank and inertias of matrix pencils generated from two/three orthogonal projections are also derived. As applications, some necessary and sufficient conditions for some matrix equalities as well as some matrix inequalities in the Loewner partial ordering to hold, are established. In the final section, the author discusses some problems for further investigation.