More results on Schur complements in Euclidean Jordan algebras

Authors:
Roman Sznajder;M. Seetharama Gowda;Melania M. Moldovan
Affiliations:
Department of Mathematics, Bowie State University, Bowie, USA 20715;Department of Mathematics and Statistics, University of Maryland, Baltimore County, Baltimore, USA 21250;Department of Mathematics, Technical University of Cluj-Napoca, Cluj-Napoca, Romania 400020
Venue:
Journal of Global Optimization
Year:
2012

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Abstract

In a recent article Gowda and Sznajder (Linear Algebra Appl 432:1553---1559, 2010) studied the concept of Schur complement in Euclidean Jordan algebras and described Schur determinantal and Haynsworth inertia formulas. In this article, we establish some more results on the Schur complement. Specifically, we prove, in the setting of Euclidean Jordan algebras, an analogue of the Crabtree-Haynsworth quotient formula and show that any Schur complement of a strictly diagonally dominant element is strictly diagonally dominant. We also introduce the concept of Schur product of a real symmetric matrix and an element of a Euclidean Jordan algebra when its Peirce decomposition with respect to a Jordan frame is given. An Oppenheim type inequality is proved in this setting.