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

# Girko's Circular Law

## See also

Eigenvalue, Random Matrix, Wigner's Semicircle Law## Explore with Wolfram|Alpha

## References

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 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.

## Referenced on Wolfram|Alpha

Girko's Circular Law## Cite this as:

Weisstein, Eric W. "Girko's Circular Law."
From *MathWorld*--A Wolfram Web Resource. https://mathworld.wolfram.com/GirkosCircularLaw.html