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

TL;DR

This paper characterizes conditions for the existence of a contractive matrix K that makes a specific non-Hermitian block matrix have a positive definite Schur complement, linking it to Krein space frame theory.

Contribution

It provides a characterization of when such matrices K exist, ensures diagonalizability, and computes the spectrum, connecting matrix theory with Krein space frames.

Findings

Existence conditions for matrix K are established.

Diagonalizability of the block matrix can be guaranteed.

Spectrum of the block matrix is explicitly computed.

Abstract

Given a positive definite matrix $A\in \mathbb{C}^{n\times n}$ and a Hermitian matrix $D\in \mathbb{C}^{m\times m}$, we characterize under which conditions there exists a strictly contractive matrix $K\in \mathbb{C}^{n\times m}$ such that the non-Hermitian block-matrix \[ \left[ \begin{array}{cc} A & -AK \\ K^*A & D \end{array} \right] \] 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.