A Monotonicity Property of the Joint Spectral Radius

We show that the joint spectral radius of a set of matrices is strictly increasing as a function of the data in the sense that if a set of matrices is contained in the relative interior of the convex hull of an irreducible set of matrices, then the joint spectral radius of the smaller set is strictly smaller than that of the larger set. This observation has some consequences in the theory of time-varying stability radii and their calculation. We show by example that, strict monotonicity notwithstanding, 0 may be a proximal normal of the joint spectral radius of some (finitely parameterized) matrix polytopes functions. This shows that the time-varying stability radius is not in general Lipschitz continuous when it is continuous