Positive diagonal solutions to the Lyapunov equations

Abstract

This document studies various stability type conditions on a matrix A related to existence of a positive diagonal matrix D such that the Lyapunov matrix Q = AD+DA(transform) is positive definite. Such problems arise in mathematical economics, in the study of time-invariant continuous-time systems and in the study of predator-prey systems. Using a theorem of the alternative, a characterization is given for all A such that a corresponding D exists. In addition, some necessary conditions for consistency and some related ideas are discussed. Finally, a method for constructing D is given for matrices A satisfying certain conditions.