A new inequality about matrix products and a Berger-Wang formula

Abstract

We prove an inequality relating the norm of a product of matrices \(A_n\cdots A_1\) with the spectral radii of subproducts \(A_j\cdots A_i\) with \(1\leq i\leq j\leq n\). Among the consequences of this inequality, we obtain the classical Berger-Wang formula as an immediate corollary, and give an easier proof of a characterization of the upper Lyapunov exponent due to I. Morris. As main ingredient for the proof of this result, we prove that for a big enough \(n\), the product \(A_n\cdots A_1\) is zero under the hypothesis that \(A_j\cdots A_i\) are nilpotent for all \(1\leq i \leq j\leq n\).