A restricted signature normal form for Hermitian matrices, quasi-spectral decompositions, and applications

AbstractIn recent years, a number of results on the relationships between the inertias of Hermitian matrices and the inertias of their principal submatrices have appeared in the literature. In this paper, we study restricted congruence transformations of Hermitian matrices M that, at the same time, induce a congruence transformation of a given principal submatrix A of M. Such transformations lead to the concept of the restricted signature normal form of M. In particular, by means of this normal form, we obtain new and shorter proofs for several known inertia theorems and also derive some new results of this type. For some applications, a special class of “almost” unitary restricted congruence transformations turns out to be useful. We show that, with such transformations, M can be reduced to a quasi-diagonal form, which, in particular, displays the eigenvalues of A. Moreover, this quasi-spectral decomposition is used to derive a generalized signature formula and to study Hermitian matrix pencils