On a class of non-Hermitian matrices with positive definite Schur complements

Abstract

Given a positive definite nXn matrix A and a Hermitian mXm matrix D, we characterize under which conditions there exists a strictly contractive matrix K such that the non-Hermitian block-matrix with the enties A and -AK in the first row and K^*A and D in the second has a positive definite Schur complement with respect to its submatrix A. Additionally, we show that K can be chosen such that diagonalizability of the block-matrix is guaranteed and we compute its spectrum. Moreover, we show a connection to the recently developed frame theory for Krein spaces.