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

This article is cited in 9 scientific papers (total in 9 papers)

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

Full text: PDF file (185 kB)
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English version:
Functional Analysis and Its Applications, 2010, 44:3, 230–233

Bibliographic databases:

UDC: 517.98+519.2
Received: 02.12.2009

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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    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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    2. Protasov V.Yu., Voynov A.S., “Sets of nonnegative matrices without positive products”, Linear Algebra Appl., 437:3 (2012), 749–765  crossref  mathscinet  zmath  isi  elib  scopus
    3. Guglielmi N., Protasov V., “Exact computation of joint spectral characteristics of linear operators”, Found. Comput. Math., 13:1 (2013), 37–97  crossref  mathscinet  zmath  isi  elib  scopus
    4. V. Yu. Protasov, “Asymptotics of Products of Nonnegative Random Matrices”, Funct. Anal. Appl., 47:2 (2013), 138–147  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    5. Voynov A., “Shortest Positive Products of Nonnegative Matrices”, Linear Alg. Appl., 439:6 (2013), 1627–1634  crossref  mathscinet  zmath  isi  elib  scopus
    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  crossref  mathscinet  zmath  isi  elib  scopus
    7. Blondel V.D., Jungers R.M., Olshevsky A., “on Primitivity of Sets of Matrices”, Automatica, 61 (2015), 80–88  crossref  mathscinet  zmath  isi  elib  scopus
    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  crossref  mathscinet  zmath  isi  elib  scopus
    9. Guglielmi N., Laglia L., Protasov V., “Polytope Lyapunov Functions For Stable and For Stabilizable Lss”, Found. Comput. Math., 17:2 (2017), 567–623  crossref  mathscinet  zmath  isi  scopus
  • Функциональный анализ и его приложения Functional Analysis and Its Applications
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