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The Perron–Frobenius theorem – a proof with the use of Markov chains

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The Perron–Frobenius theorem for an irreducible nonnegative matrix is proved using the matrix graph and the ergodic theorem of the theory of Markov chains. Bibliography: 7 titles.

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References

  1. F. R. Gantmakher, The Theory of Matrices [in Russian], Nauka, Moscow (1967).

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  4. V. Romanovsky, Un théorème sur les zéros des matrices non négatives, Bull. Soc. Math. France, 61, 213–219 (1933).

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  5. V. V. Romanovsky, Discrete Markov Chains [in Russian], Moscow (1949).

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Authors and Affiliations

  1. Kazan’ State University, Kazan’, Russia

    Yu. A. Al’pin

  2. Kazan’ State Technological University, Kazan’, Russia

    V. S. Al’pina

Authors
  1. Yu. A. Al’pin
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  2. V. S. Al’pina
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Corresponding author

Correspondence to Yu. A. Al’pin.

Additional information

Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 359, 2008, pp. 5–16.

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Al’pin, Y.A., Al’pina, V.S. The Perron–Frobenius theorem – a proof with the use of Markov chains. J Math Sci 157, 675–680 (2009). https://doi.org/10.1007/s10958-009-9347-9

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  • DOI: https://doi.org/10.1007/s10958-009-9347-9

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