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

2018/07/23 by Berger, Thomas, Giribet, Juan Ignacio, Pería, Francisco Martínez +1
#15B48 #FOS: Mathematics #Functional Analysis (math.FA) #Primary 15A83 #Secondary 15A23

paper · doi:10.48550/arxiv.1807.08591

Abstract

Given a positive definite matrix A∈ ℂ^{n× n} and a Hermitian matrix D∈ ℂ^{m× m}, we characterize under which conditions there exists a strictly contractive matrix K∈ ℂ^{n× m} such that the non-Hermitian block-matrix [ A · amp; -AK
K^*A · amp; D ] 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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