A new inequality about matrix products and a Berger-Wang formula
[Une nouvelle inégalité sur les produits de matrices et une formule de Berger-Wang]
Journal de l'École polytechnique — Mathématiques, Tome 7 (2020) , pp. 185-200.

Nous montrons une inégalité reliant la norme d’un produit ${A}_{n}\cdots {A}_{1}$ de matrices aux rayons spectraux des sous-produits ${A}_{j}\cdots {A}_{i}$ avec $1\le i\le j\le n$. 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 ${A}_{n}\cdots {A}_{1}$ est nul si les ${A}_{j}\cdots {A}_{i}$ sont nilpotents pour tout $i,j$ tel que $1\le i\le j\le n$.

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 $1\le i\le j\le n$. 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 $i,j$ such that $1\le i\le j\le n$.

Reçu le : 2018-08-22
Accepté le : 2019-11-28
Publié le : 2020-01-15
DOI : https://doi.org/10.5802/jep.114
Classification : 37H15,  15A42
Mots clés: Cocycle linéaire, rayon spectral joint, formule de Berger-Wang, exposant de Liapounov, produit de matrices nilpotentes
@article{JEP_2020__7__185_0,
author = {Eduardo Oreg\'on-Reyes},
title = {A new inequality about matrix products and a~Berger-Wang formula},
journal = {Journal de l'\'Ecole polytechnique --- Math\'ematiques},
publisher = {\'Ecole polytechnique},
volume = {7},
year = {2020},
pages = {185-200},
doi = {10.5802/jep.114},
mrnumber = {4054333},
zbl = {07152734},
language = {en},
url = {jep.centre-mersenne.org/item/JEP_2020__7__185_0/}
}
Eduardo Oregón-Reyes. A new inequality about matrix products and a Berger-Wang formula. Journal de l'École polytechnique — Mathématiques, Tome 7 (2020) , pp. 185-200. doi : 10.5802/jep.114. https://jep.centre-mersenne.org/item/JEP_2020__7__185_0/

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