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 .
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 .
Accepted:
Published online:
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
Eduardo Oregón-Reyes 1
@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{\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 = {https://jep.centre-mersenne.org/articles/10.5802/jep.114/} }
TY - JOUR AU - Eduardo Oregón-Reyes 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 - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.114/ DO - 10.5802/jep.114 LA - en ID - JEP_2020__7__185_0 ER -
%0 Journal Article %A Eduardo Oregón-Reyes %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 École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.114/ %R 10.5802/jep.114 %G en %F 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, Volume 7 (2020), pp. 185-200. doi : 10.5802/jep.114. https://jep.centre-mersenne.org/articles/10.5802/jep.114/
[1] - “A formula with some applications to the theory of Lyapunov exponents”, Israel J. Math. 131 (2002), p. 125-137 | DOI | MR | Zbl
[2] - “Bounded semigroups of matrices”, Linear Algebra Appl. 166 (1992), p. 21-27 | DOI | MR | Zbl
[3] - “Genericity of zero Lyapunov exponents”, Ergodic Theory Dynam. Systems 22 (2002) no. 6, p. 1667-1696 | DOI | MR | Zbl
[4] - “Inequalities for numerical invariants of sets of matrices”, Linear Algebra Appl. 368 (2003), p. 71-81 | DOI | MR | Zbl
[5] - “A height gap theorem for finite subsets of and nonamenable subgroups”, Ann. of Math. (2) 174 (2011) no. 2, p. 1057-1110 | DOI | MR | Zbl
[6] - Metric spaces of non-positive curvature, Grundlehren Math. Wiss., vol. 319, Springer-Verlag, Berlin, 1999 | DOI | MR | Zbl
[7] - Géométrie et théorie des groupes. Les groupes hyperboliques de Gromov, Lect. Notes in Math., vol. 1441, Springer-Verlag, Berlin, 1990 | Zbl
[8] - 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
[9] - “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) | DOI | MR | Zbl
[10] - Algebraic curves. An introduction to algebraic geometry, Advanced Book Classics, Addison-Wesley Publishing Company, Redwood City, CA, 1989 | Zbl
[11] - “Stability and Lyapunov stability of dynamical systems: a differential approach and a numerical method”, Phys. D 27 (1987) no. 3, p. 311-337 | DOI | MR | Zbl
[12] - “Subadditive and multiplicative ergodic theorems”, J. Eur. Math. Soc. (JEMS) (to appear)
[13] - “Stability of discrete linear inclusion”, Linear Algebra Appl. 231 (1995), p. 47-85 | DOI | MR | Zbl
[14] - The joint spectral radius. Theory and applications, Lect. Notes in Control and Information Sci., vol. 385, Springer-Verlag, Berlin, 2009 | DOI | MR
[15] - “A multiplicative ergodic theorem and nonpositively curved spaces”, Comm. Math. Phys. 208 (1999) no. 1, p. 107-123 | DOI | MR | Zbl
[16] - “The Berger-Wang formula for the Markovian joint spectral radius”, Linear Algebra Appl. 448 (2014), p. 315-328 | DOI | MR | Zbl
[17] - “Sharp estimates for the arithmetic Nullstellensatz”, Duke Math. J. 109 (2001) no. 3, p. 521-598 | DOI | MR | Zbl
[18] - Algebra, Graduate Texts in Math., vol. 211, Springer-Verlag, New York, 2002 | DOI | Zbl
[19] - Algebra. Vol. II: Fields with structure, algebras and advanced topics, Universitext, Springer, New York, 2008 | Zbl
[20] - “The Burnside problem for semigroups”, J. Algebra 34 (1975), p. 292-299 | DOI | MR | Zbl
[21] - “The generalised Berger-Wang formula and the spectral radius of linear cocycles”, J. Funct. Anal. 262 (2012) no. 3, p. 811-824 | DOI | MR | Zbl
[22] - “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 | DOI | MR | Zbl
[23] - “An inequality for the matrix pressure function and applications”, Adv. Math. 302 (2016), p. 280-308 | DOI | MR | Zbl
[24] - “Properties of sets of isometries of Gromov hyperbolic spaces”, Groups Geom. Dyn. 12 (2018) no. 3, p. 889-910 | DOI | MR | Zbl
[25] - Simultaneous triangularization, Universitext, Springer-Verlag, New York, 2000 | DOI | Zbl
[26] - “A note on the joint spectral radius”, Indag. Math. 22 (1960), p. 379-381 | DOI | MR | Zbl
[27] - Convex bodies: the Brunn-Minkowski theory, Encyclopedia of Math. and its Appl., vol. 151, Cambridge University Press, Cambridge, 2014 | MR | Zbl
[28] - “A sparse effective Nullstellensatz”, Adv. in Appl. Math. 22 (1999) no. 2, p. 271-295 | DOI | MR | Zbl
[29] - “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 | DOI | MR | Zbl
Cited by Sources: