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

Berger, Thomas GND; Giribet, Juan; Martínez Pería, Francisco; Trunk, Carsten GND

Given a positive definite nXn matrix A and a Hermitian mXm matrix D, we characterize under which conditions there exists a strictly contractive matrix K such that the non-Hermitian block-matrix with the enties A and -AK in the first row and K^*A and D in the second 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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Berger, Thomas / Giribet, Juan / Martínez Pería, Francisco / et al: On a class of non-Hermitian matrices with positive definite Schur complements. Ilmenau 2018.

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