A new inequality about matrix products and a Berger-Wang formula
Journal de l’École polytechnique — Mathématiques, Volume 7 (2020), pp. 185-200.

We prove an inequality relating the norm of a product of matrices A n A 1 with the spectral radii of subproducts A j A i with 1ijn. 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 A 1 is zero under the hypothesis that A j A i are nilpotent for all i,j such that 1ijn.

Nous montrons une inégalité reliant la norme d’un produit A n A 1 de matrices aux rayons spectraux des sous-produits A j A i avec 1ijn. Comme conséquences de cette inégalité, nous obtenons la formule classique de Berger-Wang comme corollaire immédiat, et nous donnons une preuve plus simple de la caractérisation, due à I. Morris, de l’exposant de Liapounov supérieur. Nous montrons, comme ingrédient principal de la preuve de ce résultat, que pour n assez grand, le produit A n A 1 est nul si les A j A i sont nilpotents pour tout i,j tel que 1ijn.

Received:
Accepted:
Published online:
DOI: 10.5802/jep.114
Classification: 37H15, 15A42
Keywords: Linear cocycle, joint spectral radius, Berger-Wang formula, Lyapunov exponent, product of nilpotent matrices
Mot clés : Cocycle linéaire, rayon spectral joint, formule de Berger-Wang, exposant de Liapounov, produit de matrices nilpotentes

Eduardo Oregón-Reyes 1

1 Department of Mathematics, University of California at Berkeley 848 Evans Hall, Berkeley, CA 94720-3860, U.S.A.
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Eduardo Oregón-Reyes. A new inequality about matrix products and a Berger-Wang formula. Journal de l’École polytechnique — Mathématiques, Volume 7 (2020), pp. 185-200. doi : 10.5802/jep.114. https://jep.centre-mersenne.org/articles/10.5802/jep.114/

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