Girko's Circular Law


Let lambda be (possibly complex) eigenvalues of a set of random n×n real matrices with entries independent and taken from a standard normal distribution. Then as n->infty, lambda/sqrt(n) is uniformly distributed on the unit disk in the complex plane. For small n, the distribution shows a concentration along the real line accompanied by a slight paucity above and below (with interesting embedded structure). However, as n->infty, the concentration about the line disappears and the distribution becomes truly uniform.

See also

Eigenvalue, Random Matrix, Wigner's Semicircle Law

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Bai, Z. D. "Circular Law." Ann. Prob. 25, 494-529, 1997.Bai, Z. D. and Yin, Y. Q. "Limiting Behavior of the Norm Products of Random Matrices and Two Problems of Geman-Hwang." Probab. Theory Related Fields 73, 555-569, 1986.Edelman, A. and Kostlan, E. "How Many Zeros of a Random Polynomial are Real?" Bull. Amer. Math. Soc. 32, 1-37, 1995.Edelman, A. "The Probability that a Random Real Gaussian Matrix has k Real Eigenvalues, Related Distributions, and the Circular Law." J. Multivariate Anal. 60, 203-232, 1997.Geman, S. "The Spectral Radius of Large Random Matrices." Ann. Probab. 14, 1318-1328, 1986.Girko, V. L. "Circular Law." Theory Probab. Appl. 29, 694-706, 1984.Girko, V. L. Theory of Random Determinants. Boston, MA: Kluwer, 1990.Mehta, M. L. Random Matrices, 3rd ed. New York: Academic Press, 2004.

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Girko's Circular Law

Cite this as:

Weisstein, Eric W. "Girko's Circular Law." From MathWorld--A Wolfram Web Resource.

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