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arXiv · 1807.08591

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

Thomas Berger·Juan Ignacio Giribet·Francisco Martínez Pería·Carsten Trunk
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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.

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Thomas Berger, Juan Ignacio Giribet, Francisco Martínez Pería, Carsten Trunk. 2018-07-23. On a class of non-Hermitian matrices with positive definite Schur complements. https://arxiv.org/abs/1807.08591

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