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 ${A}_{n}\cdots {A}_{1}$ 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 $1\le i\le j\le n$.

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 $1\le i\le j\le n$.

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
Eduardo Oregón-Reyes 1

1 Department of Mathematics, University of California at Berkeley 848 Evans Hall, Berkeley, CA 94720-3860, U.S.A.
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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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