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

Peter J. Forrester () and Santosh Kumar ()
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Peter J. Forrester: The University of Melbourne
Santosh Kumar: Shiv Nadar University

Journal of Theoretical Probability, 2018, vol. 31, issue 4, 2056-2071

Abstract: 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.

Keywords: Random matrix products; Truncated orthogonal matrices; Probability of real eigenvalues; Meijer G-functions; Arithmetic structures; 15A52; 15A15; 15A18; 33E20; 11B37 (search for similar items in EconPapers)
Date: 2018
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DOI: 10.1007/s10959-017-0766-0

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