Nous montrons une inégalité reliant la norme d’un produit de matrices aux rayons spectraux des sous-produits avec . 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 assez grand, le produit est nul si les sont nilpotents pour tout tel que .
We prove an inequality relating the norm of a product of matrices with the spectral radii of subproducts with . 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 , the product is zero under the hypothesis that are nilpotent for all such that .
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Publié le :
DOI : 10.5802/jep.114
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
@article{JEP_2020__7__185_0, author = {Oreg\'on-Reyes, Eduardo}, title = {A new inequality about matrix products and {a~Berger-Wang} formula}, journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques}, pages = {185--200}, publisher = {Ecole polytechnique}, volume = {7}, year = {2020}, doi = {10.5802/jep.114}, mrnumber = {4054333}, zbl = {07152734}, language = {en}, url = {http://www.numdam.org/articles/10.5802/jep.114/} }
TY - JOUR AU - Oregón-Reyes, Eduardo TI - A new inequality about matrix products and a Berger-Wang formula JO - Journal de l’École polytechnique — Mathématiques PY - 2020 SP - 185 EP - 200 VL - 7 PB - Ecole polytechnique UR - http://www.numdam.org/articles/10.5802/jep.114/ DO - 10.5802/jep.114 LA - en ID - JEP_2020__7__185_0 ER -
%0 Journal Article %A Oregón-Reyes, Eduardo %T A new inequality about matrix products and a Berger-Wang formula %J Journal de l’École polytechnique — Mathématiques %D 2020 %P 185-200 %V 7 %I Ecole polytechnique %U http://www.numdam.org/articles/10.5802/jep.114/ %R 10.5802/jep.114 %G en %F JEP_2020__7__185_0
Oregón-Reyes, Eduardo. 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. http://www.numdam.org/articles/10.5802/jep.114/
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