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The Probability that a Random Real Gaussian Matrix haskReal Eigenvalues, Related Distributions, and the Circular Law

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  • Edelman, Alan

Abstract

LetAbe annbynmatrix whose elements are independent random variables with standard normal distributions. Girko's (more general) circular law states that the distribution of appropriately normalized eigenvalues is asymptotically uniform in the unit disk in the complex plane. We derive the exact expected empirical spectral distribution of the complex eigenvalues for finiten, from which convergence in the expected distribution to the circular law for normally distributed matrices may be derived. Similar methodology allows us to derive a joint distribution formula for the real Schur decomposition ofA. Integration of this distribution yields the probability thatAhas exactlykreal eigenvalues. For example, we show that the probability thatAhas all real eigenvalues is exactly 2-n(n-1)/4.

Suggested Citation

  • Edelman, Alan, 1997. "The Probability that a Random Real Gaussian Matrix haskReal Eigenvalues, Related Distributions, and the Circular Law," Journal of Multivariate Analysis, Elsevier, vol. 60(2), pages 203-232, February.
  • Handle: RePEc:eee:jmvana:v:60:y:1997:i:2:p:203-232
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    Cited by:

    1. Pan, Guangming & Zhou, Wang, 2010. "Circular law, extreme singular values and potential theory," Journal of Multivariate Analysis, Elsevier, vol. 101(3), pages 645-656, March.
    2. Chafaï, Djalil, 2010. "The Dirichlet Markov Ensemble," Journal of Multivariate Analysis, Elsevier, vol. 101(3), pages 555-567, March.
    3. Causeur, David & Dhorne, Thierry & Antoni, Arlette, 2005. "A two-way analysis of variance model with positive definite interaction for homologous factors," Journal of Multivariate Analysis, Elsevier, vol. 95(2), pages 431-448, August.

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