On the accuracy of the ellipsoid norm approximation of the joint spectral radius

Abstract

The joint spectral radius of a set of matrices is a measure of the maximal asymptotic growth rate that can be obtained by forming long products of matrices taken from the set. This quantity appears in a number of application contexts but is notoriously difficult to compute and to approximate. We introduce in this paper an approximation ρ ˆ that is based on ellipsoid norms, that can be computed by convex optimization, and that is such that the joint spectral radius belongs to the interval [ ρ ˆ / n , ρ ˆ ] , where n is the dimension of the matrices. We also provide a simple approximation for the special case where the entries of the matrices are non-negative; in this case the approximation is proved to be within a factor at most m ( m is the number of matrices) of the exact value.