V. Yu. Protasov, “Invariant Functionals for Random Matrices”, Funktsional. Anal. i Prilozhen., 44:3 (2010), 84–88; Funct. Anal. Appl., 44:3 (2010), 230

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Funktsional'nyi Analiz i ego Prilozheniya, 2010, Volume 44, Issue 3, Pages 84–88
DOI: https://doi.org/10.4213/faa3002 (Mi faa3002)

This article is cited in 19 scientific papers (total in 19 papers)

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Invariant Functionals for Random Matrices

V. Yu. Protasov

Moscow State University

Full-text PDF (185 kB) Citations (19)

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DOI: https://doi.org/10.4213/faa3002

Abstract: A new approach to the study of the Lyapunov exponents of random matrices is presented. It is proved that, under general assumptions, any family of nonnegative matrices possesses a continuous concave positively homogeneous invariant functional (“antinorm”) on $\mathbb{R}^d_+$. Moreover, the coefficient corresponding to an invariant antinorm equals the largest Lyapunov exponent. All conditions imposed on the matrices are shown to be essential. As a corollary, a sharp estimate for the asymptotics of the mathematical expectation for logarithms of norms of matrix products and of their spectral radii is derived. New upper and lower bounds for Lyapunov exponents are obtained. This leads to an algorithm for computing Lyapunov exponents. The proofs of the main results are outlined.

Keywords: random matrices, Lyapunov exponents, invariant functions, concave homogeneous functionals, fixed point, asymptotics.

Received: 02.12.2009

English version:
Functional Analysis and Its Applications, 2010, Volume 44, Issue 3, Pages 230–233
DOI: https://doi.org/10.1007/s10688-010-0031-0

Bibliographic databases:

Document Type: Article

UDC: 517.98+519.2

Language: Russian

Citation: V. Yu. Protasov, “Invariant Functionals for Random Matrices”, Funktsional. Anal. i Prilozhen., 44:3 (2010), 84–88; Funct. Anal. Appl., 44:3 (2010), 230–233

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This publication is cited in the following 19 articles:

M. S. Makarov, V. Yu. Protasov, “Avtopolyarnye konicheskie tela i mnogogranniki”, Matem. sb., 216:3 (2025), 156–176
M. S. Makarov, “Antinormy i avtopolyarnye mnogogranniki”, Sib. matem. zhurn., 64:5 (2023), 1050–1064
Yaokun Wu, Yinfeng Zhu, “Primitivity and Hurwitz Primitivity of Nonnegative Matrix Tuples: A Unified Approach”, SIAM J. Matrix Anal. Appl., 44:1 (2023), 196
M. S. Makarov, “Antinorms and Self-Polar Polyhedra”, Sib Math J, 64:5 (2023), 1200
Matteo Della Rossa, Raphael M. Jungers, 2022 IEEE 61st Conference on Decision and Control (CDC), 2022, 1021
Protasov V.Yu., “Antinorms on Cones: Duality and Applications”, Linear Multilinear Algebra, 2021
Guglielmi N., Zennaro M., “An Antinorm Theory For Sets of Matrices: Bounds and Approximations to the Lower Spectral Radius”, Linear Alg. Appl., 607 (2020), 89–117
Guglielmi N., Laglia L., Protasov V., “Polytope Lyapunov Functions For Stable and For Stabilizable Lss”, Found. Comput. Math., 17:2 (2017), 567–623
Nicola Guglielmi, Vladimir Protasov, AIP Conference Proceedings, 1738, 2016, 020006
Blondel V.D., Jungers R.M., Olshevsky A., “on Primitivity of Sets of Matrices”, Automatica, 61 (2015), 80–88
Guglielmi N., Zennaro M., “Canonical Construction of Polytope Barabanov Norms and Antinorms For Sets of Matrices”, SIAM J. Matrix Anal. Appl., 36:2 (2015), 634–655
Guglielmi N., Protasov V., “Exact computation of joint spectral characteristics of linear operators”, Found. Comput. Math., 13:1 (2013), 37–97
V. Yu. Protasov, “Asymptotics of Products of Nonnegative Random Matrices”, Funct. Anal. Appl., 47:2 (2013), 138–147
Voynov A., “Shortest Positive Products of Nonnegative Matrices”, Linear Alg. Appl., 439:6 (2013), 1627–1634
Protasov V.Yu., Jungers R.M., “Lower and Upper Bounds for the Largest Lyapunov Exponent of Matrices”, Linear Alg. Appl., 438:11 (2013), 4448–4468
Vincent D. Blondel, Raphael M. Jungers, Alex Olshevsky, 52nd IEEE Conference on Decision and Control, 2013, 1360
Nicola Guglielmi, Linda Laglia, 52nd IEEE Conference on Decision and Control, 2013, 710
Protasov V.Yu., Voynov A.S., “Sets of nonnegative matrices without positive products”, Linear Algebra Appl., 437:3 (2012), 749–765
V. Yu. Protasov, “Semigroups of non-negative matrices”, Russian Math. Surveys, 65:6 (2010), 1186–1188

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

Functional Analysis and Its Applications

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