A new sufficient condition for the uniqueness of Barabanov norms

TL;DR

This paper introduces a new sufficient condition for the uniqueness of Barabanov norms related to the joint spectral radius of matrix sets, highlighting the sensitivity of this property to small perturbations.

Contribution

It provides a novel sufficient condition for the uniqueness of Barabanov norms and demonstrates its applicability through examples and theoretical analysis.

Findings

New sufficient condition for Barabanov norm uniqueness

Examples where the condition applies

Sensitivity of the uniqueness property to perturbations

Abstract

The joint spectral radius of a bounded set of d times d real or complex matrices is defined to be the maximum exponential rate of growth of products of matrices drawn from that set. Under quite mild conditions such a set of matrices admits an associated vector norm, called a Barabanov norm, which can be used to characterise those sequences of matrices which achieve this maximum rate of exponential growth. In this note we continue an earlier investigation into the problem of determining when the Barabanov norm associated to such a set of matrices is unique. We give a new sufficient condition for this uniqueness, and provide some examples in which our condition applies. We also give a theoretical application which shows that the property of having a unique Barabanov norm can in some cases be highly sensitive to small perturbations of the set of matrices.