On the stability of a convex set of matrices (Q2369054)

It is well-known that the stability of a compact set of complex matrices may be characterized through a common solution of directional Lyapunov inclusions. In this paper the authors give an alternative proof of this result. It is shown that the companion matrix whose non-trivial column is negative satisfies the directional Lyapunov condition (inclusion) for real multiplier vectors. Using a minimax approach, a stability criterion for subsets of \(Z\)-matrices, based on real vectors and positive definite diagonal matrices, is provided.