Complementarity properties of the Lyapunov transformation over symmetric cones
type: Item
instance of: scholarly article
title: Complementarity properties of the Lyapunov transformation over symmetric cones
zbMATH Open document ID: 1264.90168
publication date: 2012-11-07T00:00:00Z
review text: From the text: ``The well-known Lyapunov's theorem in matrix theory/continuous dynamical systems asserts that a square matrix \(A\) is positive stable if and only if there exists a positive definite matrix \(X\) such that \(AX+XA*\) is positive definite. Motivated by this theorem and the complementarity form of this theorem, in this paper, we study some complementarity properties of the Lyapunov transformation defined on any Euclidean Jordan algebra \(V\). Given any element \(a \in V\), we consider the corresponding Lyapunov transformation \(L_a\) and show that the \(P\) and \(S\)-properties are both equivalent to a being positive. Then we characterize the \(R_0\)-property for \(L_a\) and show that \(L_a\) has the \(R_0\)-property if and only if a is invertible. Finally, we provide \(L_a\) with some characterizations of the \(E_0\)-property and the nondegeneracy property. In order to make our results clearly, we summarize the implications between these complementarity properties in a diagram for the Lyapunov transformation.''
reviewed by: Olaf Ninnemann
Mathematics Subject Classification ID: 90C33
Mathematics Subject Classification ID: 17C55
zbMATH DE Number: 6102800
zbMATH Keywords: Euclidean Jordan algebra
zbMATH Keywords: Lyapunov transformation
zbMATH Keywords: symmetric cone complementarity problem
author: Yuan Min Li
author: Xing Tao Wang
author: Deyun Wei
describes a project that uses: SCCP
MaRDI profile type: Publication
full work available at URL: s10114-012-0138-8
OpenAlex ID: W2158633927
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DOI: 10.1007/S10114-012-0138-8
published in: Acta Mathematica Sinica, English Series
description: scientific article; zbMATH DE number 6102800