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

TL;DR

This paper introduces a new inequality connecting matrix product norms with spectral radii of subproducts, leading to a simplified proof of the Berger-Wang formula and related results on Lyapunov exponents.

Contribution

It establishes a novel inequality that unifies and simplifies proofs of key results in matrix theory and dynamical systems.

Findings

Proves a new inequality relating matrix product norms and spectral radii.

Derives the Berger-Wang formula as a corollary.

Provides an easier proof of a characterization of the upper Lyapunov exponent.

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 large enough $n$, the product $A_n\cdots A_1$ is zero under the hypothesis that $A_j\cdots A_i$ are nilpotent for all $1\leq i \leq j\leq n$.