Journal of Theoretical Probability

, Volume 31, Issue 4, pp 2056–2071 | Cite as

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

  • Peter J. Forrester
  • Santosh KumarEmail author


The probability that all eigenvalues of a product of m independent \(N \times N\) subblocks of a Haar distributed random real orthogonal matrix of size \((L_i+N) \times (L_i+N)\), \((i=1,\dots ,m)\) 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 \(L_i\) even the probability is a rational number. The formulae furthermore provide for explicit computation in small order cases.


Random matrix products Truncated orthogonal matrices Probability of real eigenvalues Meijer G-functions Arithmetic structures 

Mathematics Subject Classification (2010)

15A52 15A15 15A18 33E20 11B37 


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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.School of Mathematics and Statistics, ARC Centre of Excellence for Mathematical and Statistical FrontiersThe University of MelbourneVictoriaAustralia
  2. 2.Department of PhysicsShiv Nadar UniversityGautam Buddha NagarIndia

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