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

2017/10/02 by Oregón-Reyes, Eduardo · 1 citation
#15A42 #37H15 #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1710.00639

Abstract

We prove an inequality relating the norm of a product of matrices A_n⋯ A₁ with the spectral radii of subproducts A_j⋯ A_i with 1≤ i≤ j≤ 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⋯ A₁ is zero under the hypothesis that A_j⋯ A_i are nilpotent for all 1≤ i ≤ j≤ n.

Cited by

• Asymptotic behavior of the spectral radius of locally constant strongly irreducible cocycles

Related

• Computation of Lyapunov exponents of matrix products
• Lyapunov exponents for products of matrices
• Bounds on norms of compound matrices and on products of eigenvalues
• Equality of Lyapunov and stability exponents for products of isotropic random matrices
• An inequality for the trace of matrix products, using absolute values