Applications of an inequality for the Schur complement

Abstract

The inertia of an Hermitian matrix A is given by the ordered triplet, In A = (7r, zv, 3), where ir denotes the number of positive, v the number of negative, and a the number of zero roots of the matrix A. In a previous paper [2 ], it was shown that the inertia of an Hermitian matrix can be determined from that of any nonsingular principal submatrix together with that of its Schur complement. That is, if A is Hermitian, and B is a nonsingular principal submatrix of A, then