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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. Peter J. Forrester & Santosh Kumar, 2018. "The Probability That All Eigenvalues are Real for Products of Truncated Real Orthogonal Random Matrices," Journal of Theoretical Probability, Springer, vol. 31(4), pages 2056-2071, December.
    2. 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.
    3. Christoly Biely & Stefan Thurner, 2008. "Random matrix ensembles of time-lagged correlation matrices: derivation of eigenvalue spectra and analysis of financial time-series," Quantitative Finance, Taylor & Francis Journals, vol. 8(7), pages 705-722.
    4. Natalie Coston & Sean O’Rourke, 2020. "Gaussian Fluctuations for Linear Eigenvalue Statistics of Products of Independent iid Random Matrices," Journal of Theoretical Probability, Springer, vol. 33(3), pages 1541-1612, September.
    5. Tiefeng Jiang & Yongcheng Qi, 2017. "Spectral Radii of Large Non-Hermitian Random Matrices," Journal of Theoretical Probability, Springer, vol. 30(1), pages 326-364, March.
    6. Chafaï, Djalil, 2010. "The Dirichlet Markov Ensemble," Journal of Multivariate Analysis, Elsevier, vol. 101(3), pages 555-567, March.
    7. Djalil Chafaï, 2010. "Circular Law for Noncentral Random Matrices," Journal of Theoretical Probability, Springer, vol. 23(4), pages 945-950, December.
    8. 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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