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 Funktsional. Anal. i Prilozhen., 2010, Volume 44, Issue 3, Pages 84–88 (Mi faa3002)

Brief communications

Invariant Functionals for Random Matrices

V. Yu. Protasov

Moscow State University

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.

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

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English version:
Functional Analysis and Its Applications, 2010, 44:3, 230–233

Bibliographic databases:

UDC: 517.98+519.2

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

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/faa3002
• https://doi.org/10.4213/faa3002
• http://mi.mathnet.ru/eng/faa/v44/i3/p84

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

This publication is cited in the following articles:
1. V. Yu. Protasov, “Semigroups of non-negative matrices”, Russian Math. Surveys, 65:6 (2010), 1186–1188
2. Protasov V.Yu., Voynov A.S., “Sets of nonnegative matrices without positive products”, Linear Algebra Appl., 437:3 (2012), 749–765
3. Guglielmi N., Protasov V., “Exact computation of joint spectral characteristics of linear operators”, Found. Comput. Math., 13:1 (2013), 37–97
4. V. Yu. Protasov, “Asymptotics of Products of Nonnegative Random Matrices”, Funct. Anal. Appl., 47:2 (2013), 138–147
5. Voynov A., “Shortest Positive Products of Nonnegative Matrices”, Linear Alg. Appl., 439:6 (2013), 1627–1634
6. 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
7. Blondel V.D., Jungers R.M., Olshevsky A., “on Primitivity of Sets of Matrices”, Automatica, 61 (2015), 80–88
8. 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
9. Guglielmi N., Laglia L., Protasov V., “Polytope Lyapunov Functions For Stable and For Stabilizable Lss”, Found. Comput. Math., 17:2 (2017), 567–623
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