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The Probability That All Eigenvalues are Real for Products of Truncated Real Orthogonal Random Matrices

Author

Listed:
  • Peter J. Forrester

    (The University of Melbourne)

  • Santosh Kumar

    (Shiv Nadar University)

Abstract

The probability that all eigenvalues of a product of m independent $$N \times N$$ N × N subblocks of a Haar distributed random real orthogonal matrix of size $$(L_i+N) \times (L_i+N)$$ ( L i + N ) × ( L i + N ) , $$(i=1,\dots ,m)$$ ( i = 1 , ⋯ , m ) are real is calculated as a multidimensional integral, and as a determinant. Both involve Meijer G-functions. Evaluation formulae of the latter, based on a recursive scheme, allow it to be proved that for any m and with each $$L_i$$ L i even the probability is a rational number. The formulae furthermore provide for explicit computation in small order cases.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jotpro:v:31:y:2018:i:4:d:10.1007_s10959-017-0766-0
    DOI: 10.1007/s10959-017-0766-0
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    References listed on IDEAS

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    1. Jos Berge, 1991. "Kruskal's polynomial for 2×2×2 arrays and a generalization to 2×n×n arrays," Psychometrika, Springer;The Psychometric Society, vol. 56(4), pages 631-636, December.
    2. 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.
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