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The Subdominant Eigenvalue of a Large Stochastic Matrix

Author

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  • Gyorgy Molnar
  • Andras Simonovits

Abstract

Using intuition and computer experimentation, Brady conjectured that the ratio of the subdominant eigenvalue to the dominant eigenvalue of a positive random matrix (with identically and independently distributed entries) converges to zero when the number of the sectors tends to infinity. In this paper, we discuss the deterministic case and, among other things, prove the following version of this conjecture: if each entry of the matrix deviates from 1/n by at most θ/n1+е, then the modulus of the subdominant root is at most θ/nе where θ and ε are arbitrary positive real parameters.

Suggested Citation

  • Gyorgy Molnar & Andras Simonovits, 1998. "The Subdominant Eigenvalue of a Large Stochastic Matrix," Economic Systems Research, Taylor & Francis Journals, vol. 10(1), pages 79-82.
    • Handle: RePEc:taf:ecsysr:v:10:y:1998:i:1:p:79-82
    DOI: 10.1080/09535319800000007
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    Cited by:

    1. Theodore Mariolis & Lefteris Tsoulfidis, 2012. "On Brody’S Conjecture: Facts And Figures From The Us Economy," Discussion Paper Series 2012_06, Department of Economics, University of Macedonia, revised May 2012.
    2. Gurgul, Henryk, 2007. "Stochastic input-output modeling," MPRA Paper 68573, University Library of Munich, Germany, revised 2007.
    3. Christian Bidard & Tom Schatteman, 2001. "The Spectrum of Random Matrices," Economic Systems Research, Taylor & Francis Journals, vol. 13(3), pages 289-298.
    4. Anwar Shaikh & Luiza Nassif, 2018. "Eigenvalue distribution, matrix size and the linearity of wage-profit curves," Working Papers 1812, New School for Social Research, Department of Economics.
    5. Andras Brody, 2000. "The Monetary Multiplier," Economic Systems Research, Taylor & Francis Journals, vol. 12(2), pages 215-219.

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