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

Abstract:

Given Hermitian matrices $A\in \mathbb {C}^{n\times n}$ and $D\in \mathbb {C}^{m\times m}$, and $\kappa >0$, we characterize under which conditions there exists a matrix $K\in \mathbb {C}^{n\times m}$ with $\|K\|<\kappa$ such that the non-Hermitian block-matrix \begin{equation*}{\left [\begin {array}{cc} {A}&{-AK}\\ {K^*A} & {D} \end{array} \right ]} \end{equation*} has a positive (semi)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.