Notes on Simplifying the Construction of Barabanov Norms

Abstract

To answer the question about the growth rate of matrix products, the concepts of joint and generalized spectral radius were introduced in the 1960s. A common tool for finding the joint/generalized spectral radius is the so-called extremal norms and, in particular, the Barabanov norm. The goal of this paper is to try to combine the advantages of different approaches based on the concept of extremality in order to obtain results that are simpler for everyday use. It is shown how the Dranishnikov-Konyagin theorem on the existence of a special invariant body for a set of matrices can be used to construct a Barabanov norm. A modified max-relaxation algorithm for constructing Barabanov norms, which follows from this theorem, is described. Additional techniques are also described that simplify the construction of Barabanov norms under the assumption that