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

Oregón-Reyes, Eduardo

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

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

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