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The Spectrum of Random Matrices

Author

Listed:
  • Christian Bidard
  • Tom Schatteman

Abstract

The Frobenius eigenvector of a positive square matrix is obtained by iterating the multiplication of an arbitrary positive vector by the matrix. Brody (1997) noticed that, when the entries of the matrix are independently and identically distributed, the speed of convergence increases statistically with the dimension of the matrix. As the speed depends on the ratio between the subdominant and the dominant eigenvalues, Brody's conjecture amounts to stating that this ratio tends to zero when the dimension tends to infinity. The paper provides a simple proof of the result. Some mathematical and economic aspects of the problem are discussed.

Suggested Citation

  • Christian Bidard & Tom Schatteman, 2001. "The Spectrum of Random Matrices," Economic Systems Research, Taylor & Francis Journals, vol. 13(3), pages 289-298.
    • Handle: RePEc:taf:ecsysr:v:13:y:2001:i:3:p:289-298
    DOI: 10.1080/09537320120070167
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    References listed on IDEAS

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    1. 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.
    2. Dietzenbacher, Erik, 1993. "A limiting property for the powers of a non-negative, reducible matrix," Structural Change and Economic Dynamics, Elsevier, vol. 4(2), pages 353-366, December.
    3. Stanisław Białas & Henryk Gurgul, 1998. "On Hypothesis about the Second Eigenvalue of the Leontief Matrix," Economic Systems Research, Taylor & Francis Journals, vol. 10(3), pages 285-290.
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    Citations

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    Cited by:

    1. Luis Daniel Torres Gonzalez & Jangho Yang, 2018. "The Persistent Statistical Structure of the US Input-Output Coefficient Matrices: 1963-2007," Working Papers 1804, New School for Social Research, Department of Economics.
    2. 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.
    3. Luis Daniel Torres Gonzalez, 2017. "Regularities in Prices of Production and the Concentration of Compositions of Capitals," Working Papers 1709, New School for Social Research, Department of Economics.
    4. Gurgul, Henryk, 2007. "Stochastic input-output modeling," MPRA Paper 68573, University Library of Munich, Germany, revised 2007.
    5. Luis Daniel Torres-González, 2020. "The Characteristics of the Productive Structure Behind the Empirical Regularities in Production Prices Curves," Working Papers 2016, New School for Social Research, Department of Economics.
    6. 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.
    7. Torres-González, Luis Daniel, 2022. "The Characteristics of the Productive Structure Behind the Empirical Regularities in Production Prices Curves," Structural Change and Economic Dynamics, Elsevier, vol. 62(C), pages 622-659.

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