Complementarity properties of the Lyapunov transformation over symmetric cones

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. In this paper, we extend this theorem to the setting of any Euclidean Jordan algebra V. Given any element a ∈ V, we consider the corresponding Lyapunov transformation La and show that the P and S-properties are both equivalent to a being positive. Then we characterize the R0-property for La and show that La has the R0-property if and only if a is invertible. Finally, we provide La with some characterizations of the E0-property and the nondegeneracy property.