The generalized spectral radius is strictly increasing Original Research Article

Using a result linking convexity and irreducibility of matrix sets it is shown that the generalized spectral radius of a compact set of matrices is a strictly increasing function of the set in a very natural sense. As an application some consequences of this property in the area of time-varying stability radii are discussed. In particular, using the implicit function theorem sufficient conditions for Lipschitz continuity are derived. An example is presented of a linearly increasing family of matrix polytopes for which the proximal subgradient of the generalized spectral radius at a certain polytope contains 0, so that the implicit function theorem is not applicable in all cases.