Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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


Authors: Thomas Berger, Juan Giribet, Francisco Martínez Pería and Carsten Trunk
Journal: Proc. Amer. Math. Soc. 147 (2019), 2375-2388
MSC (2010): Primary 15A83; Secondary 15A23, 15B48
DOI: https://doi.org/10.1090/proc/14412
Published electronically: March 7, 2019
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 $ \Vert K\Vert<\kappa $ such that the non-Hermitian block-matrix

    $\displaystyle {\left [\begin {array}{cc} {A}&{-AK}\\ {K^*A} & {D} \end{array} \right ]}$

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.

References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 15A83, 15A23, 15B48

Retrieve articles in all journals with MSC (2010): 15A83, 15A23, 15B48


Additional Information

Thomas Berger
Affiliation: Institut für Mathematik, Universität Paderborn, Warburger Str. 100, 33098 Paderborn, Germany
Email: thomas.berger@math.upb.de

Juan Giribet
Affiliation: Departamento de Ingeniería Electrónica y Matemática, Universidad de Buenos Aires and Instituto Argentino de Matemática “Alberto P. Calderón” (CONICET), Saavedra 15 (1083) Buenos Aires, Argentina
Email: jgiribet@fi.uba.ar

Francisco Martínez Pería
Affiliation: Centro de Matemática de La Plata (CeMaLP) – FCE-UNLP, La Plata, Argentina – and – Instituto Argentino de Matemática “Alberto P. Calderón” (CONICET), Saavedra 15 (1083) Buenos Aires, Argentina
Email: francisco@mate.unlp.edu.ar

Carsten Trunk
Affiliation: Institut für Mathematik, Technische Universität Ilmenau, Postfach 100565, D-98684 Ilmenau, Germany – and – Instituto Argentino de Matemática “Alberto P. Calderón” (CONICET), Saavedra 15 (1083) Buenos Aires, Argentina
Email: carsten.trunk@tu-ilmenau.de

DOI: https://doi.org/10.1090/proc/14412
Received by editor(s): July 17, 2018
Received by editor(s) in revised form: September 24, 2018, September 25, 2018, and October 1, 2018
Published electronically: March 7, 2019
Communicated by: Stephan Garcia
Article copyright: © Copyright 2019 American Mathematical Society