Classes of discrete linear systems having common quadratic Lyapunov functions

This paper addresses the common Lyapunov function problem for discrete-time linear systems. Namely, the authors raise the question that in order for a given set of Schur stable system matrices to share a quadratic Lyapunov function what kind of properties are required for the set. The authors provide a partial answer to it by showing that if the matrices form a commuting family then they have the desired property