The Spectrum of Random Matrices

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The Spectrum of Random Matrices

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Christian Bidard, Tom Schatteman

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Christian Bidard

Abstract

The Frobenius eigenvector of a positive square matrix is obtained by iterating the multiplication of an arbitrary positive vector by the matrix. Bródy (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, Bródy'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.

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Article provided by Taylor and Francis Journals in its journal Economic Systems Research.

Volume (Year): 13 (2001)
Issue (Month): 3 (September)
Pages: 289-298
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Handle: RePEc:taf:ecsysr:v:13:y:2001:i:3:p:289-298

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References listed on IDEAS
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:

Bialas, Stanislaw & Gurgul, Henryk, 1998. "On Hypothesis about the Second Eigenvalue of the Leontief Matrix," Economic Systems Research, Taylor and Francis Journals, vol. 10(3), pages 285-89, September.
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. [Downloadable!] (restricted)
Molnar, Gyorgy & Simonovits, Andras, 1998. "The Subdominant Eigenvalue of a Large Stochastic Matrix," Economic Systems Research, Taylor and Francis Journals, vol. 10(1), pages 79-82, March.
Brody, Andras, 1997. "The Second Eigenvalue of the Leontief Matrix," Economic Systems Research, Taylor and Francis Journals, vol. 9(3), pages 253-58, September.

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