On the joint spectral radius

Abstract

For a bounded subset S of d x d complex matrices, the Berger-Wang theorem and Bochi's inequality allow to approximate the joint spectral radius of S from below by the spectral radius of a short product of elements from S. Our goal is twofold: we review these results, providing self-contained proofs, and we derive an improved version with explicit bounds that are polynomial in d. We also discuss other complete valued fields.