Complementarity properties of the Lyapunov transformation over symmetric cones
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.''
scientific article; zbMATH DE number 6102800
scientific article