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

Abstract

Given Hermitian matrices A ∈ C n × n A\in \mathbb {C}^{n\times n} and D ∈ C m × m D\in \mathbb {C}^{m\times m} , and κ > 0 \kappa >0 , we characterize under which conditions there exists a matrix K ∈ C n × m K\in \mathbb {C}^{n\times m} with ‖ K ‖ > κ \|K\|>\kappa such that the non-Hermitian block-matrix [ A a m p ; − A K K ∗ A a m p ; D ] \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 A . Additionally, we show that  K 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.