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 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.

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.

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/

[1] A. Avila & J. Bochi - “A formula with some applications to the theory of Lyapunov exponents”, Israel J. Math. 131 (2002), p. 125-137 | Article | MR 1942304 | Zbl 1022.37019

[2] M. A. Berger & Y. Wang - “Bounded semigroups of matrices”, Linear Algebra Appl. 166 (1992), p. 21-27 | Article | MR 1152485 | Zbl 0818.15006

[3] J. Bochi - “Genericity of zero Lyapunov exponents”, Ergodic Theory Dynam. Systems 22 (2002) no. 6, p. 1667-1696 | Article | MR 1944399 | Zbl 1023.37006

[4] J. Bochi - “Inequalities for numerical invariants of sets of matrices”, Linear Algebra Appl. 368 (2003), p. 71-81 | Article | MR 1983195 | Zbl 1031.15023

[5] E. Breuillard - “A height gap theorem for finite subsets of GL d ( ¯) and nonamenable subgroups”, Ann. of Math. (2) 174 (2011) no. 2, p. 1057-1110 | Article | MR 2831113 | Zbl 1243.11071

[6] M. R. Bridson & A. Haefliger - Metric spaces of non-positive curvature, Grundlehren Math. Wiss., vol. 319, Springer-Verlag, Berlin, 1999 | Article | MR 1744486 | Zbl 0988.53001

[7] M. Coornaert, T. Delzant & A. Papadopoulos - Géométrie et théorie des groupes. Les groupes hyperboliques de Gromov, Lect. Notes in Math., vol. 1441, Springer-Verlag, Berlin, 1990 | Zbl 0727.20018

[8] T. Das, D. Simmons & M. Urbański - Geometry and dynamics in Gromov hyperbolic metric spaces. With an emphasis on non-proper settings, Math. Surveys and Monographs, vol. 218, American Mathematical Society, Providence, RI, 2017 | Zbl 06729361

[9] L. Elsner - “The generalized spectral-radius theorem: an analytic-geometric proof”, Linear Algebra Appl. 220 (1995), p. 151-159, Proceedings of the Workshop “Nonnegative Matrices, Applications and Generalizations” and the Eighth Haifa Matrix Theory Conference (Haifa, 1993) | Article | MR 1334574 | Zbl 0828.15006

[10] W. Fulton - Algebraic curves. An introduction to algebraic geometry, Advanced Book Classics, Addison-Wesley Publishing Company, Redwood City, CA, 1989 | Zbl 0681.14011

[11] I. Goldhirsch, P.-L. Sulem & S. A. Orszag - “Stability and Lyapunov stability of dynamical systems: a differential approach and a numerical method”, Phys. D 27 (1987) no. 3, p. 311-337 | Article | MR 913683 | Zbl 0656.34045

[12] S. Gouëzel & A. Karlsson - “Subadditive and multiplicative ergodic theorems”, J. Eur. Math. Soc. (JEMS) (to appear)

[13] L. Gurvits - “Stability of discrete linear inclusion”, Linear Algebra Appl. 231 (1995), p. 47-85 | Article | MR 1361100 | Zbl 0845.68067

[14] R. Jungers - The joint spectral radius. Theory and applications, Lect. Notes in Control and Information Sci., vol. 385, Springer-Verlag, Berlin, 2009 | Article | MR 2507938

[15] A. Karlsson & G. A. Margulis - “A multiplicative ergodic theorem and nonpositively curved spaces”, Comm. Math. Phys. 208 (1999) no. 1, p. 107-123 | Article | MR 1729880 | Zbl 0979.37006

[16] V. Kozyakin - “The Berger-Wang formula for the Markovian joint spectral radius”, Linear Algebra Appl. 448 (2014), p. 315-328 | Article | MR 3182989 | Zbl 1321.15021

[17] T. Krick, L. M. Pardo & M. Sombra - “Sharp estimates for the arithmetic Nullstellensatz”, Duke Math. J. 109 (2001) no. 3, p. 521-598 | Article | MR 1853355 | Zbl 1010.11035

[18] S. Lang - Algebra, Graduate Texts in Math., vol. 211, Springer-Verlag, New York, 2002 | Article | Zbl 0984.00001

[19] F. Lorenz - Algebra. Vol. II: Fields with structure, algebras and advanced topics, Universitext, Springer, New York, 2008 | Zbl 1130.12001

[20] R. McNaughton & Y. Zalcstein - “The Burnside problem for semigroups”, J. Algebra 34 (1975), p. 292-299 | Article | MR 374301 | Zbl 0302.20054

[21] I. D. Morris - “The generalised Berger-Wang formula and the spectral radius of linear cocycles”, J. Funct. Anal. 262 (2012) no. 3, p. 811-824 | Article | MR 2863849 | Zbl 1254.47006

[22] I. D. Morris - “Mather sets for sequences of matrices and applications to the study of joint spectral radii”, Proc. London Math. Soc. (3) 107 (2013) no. 1, p. 121-150 | Article | MR 3083190 | Zbl 1277.15009

[23] I. D. Morris - “An inequality for the matrix pressure function and applications”, Adv. Math. 302 (2016), p. 280-308 | Article | MR 3545931 | Zbl 1350.15005

[24] E. Oregón-Reyes - “Properties of sets of isometries of Gromov hyperbolic spaces”, Groups Geom. Dyn. 12 (2018) no. 3, p. 889-910 | Article | MR 3844998 | Zbl 1398.53049

[25] H. Radjavi & P. Rosenthal - Simultaneous triangularization, Universitext, Springer-Verlag, New York, 2000 | Article | Zbl 0981.15007

[26] G.-C. Rota & G. Strang - “A note on the joint spectral radius”, Indag. Math. 22 (1960), p. 379-381 | Article | MR 147922 | Zbl 0095.09701

[27] R. Schneider - Convex bodies: the Brunn-Minkowski theory, Encyclopedia of Math. and its Appl., vol. 151, Cambridge University Press, Cambridge, 2014 | MR 3155183 | Zbl 1287.52001

[28] M. Sombra - “A sparse effective Nullstellensatz”, Adv. in Appl. Math. 22 (1999) no. 2, p. 271-295 | Article | MR 1659402 | Zbl 0933.14001

[29] J. N. Tsitsiklis & V. D. Blondel - “The Lyapunov exponent and joint spectral radius of pairs of matrices are hard—when not impossible—to compute and to approximate”, Math. Control Signals Systems 10 (1997) no. 1, p. 31-40, Correction: Ibid., no. 4, p. 381 | Article | MR 1462278 | Zbl 0888.65044