Complex Hermitian random matrices with a unitary symmetry can be distinguished by a weight function. When this is even, it is a known result that the distribution of the singular values can be decomposed as the superposition of two independent eigenvalue sequences distributed according to particular matrix ensembles with chiral unitary symmetry. We give decompositions of the distribution of singular values, and the decimation of the singular values – whereby only even, or odd, labels are observed – for real symmetric random matrices with an orthogonal symmetry, and even weight. This requires further specifying the functional form of the weight to one of three types – Gauss, symmetric Jacobi or Cauchy. Inter-relations between gap probabilities with orthogonal and unitary symmetry follow as a corollary. The Gauss case has appeared in a recent work of Bornemann and La Croix. The Cauchy case, when appropriately specialised and upon stereographic projection, gives decompositions for the analogue of the singular values for the circular unitary and circular orthogonal ensembles.
Funding source: Deutsche Forschungsgemeinschaft
Award Identifier / Grant number: TRR 109
Funding source: Australian Research Council
Award Identifier / Grant number: DP140102613
Funding statement: The work of FB was supported by the DFG-Collaborative Research Center, TRR 109, “Discretization in Geometry and Dynamics”. The work of PJF was supported by the Australian Research Council through the grant DP140102613.
 Bornemann F., On the numerical evaluation of distributions in random matrix theory: A review, Markov Process. Related Fields 16 (2010), no. 4, 803–866. Search in Google Scholar
 de Bruijn N. G., On some multiple integrals involving determinants, J. Indian Math. Soc. (N.S.) 19 (1955), 133–151. Search in Google Scholar
 Forrester P. J. and Rains E. M., Interrelationships between orthogonal, unitary and symplectic matrix ensembles, Random Matrix Models and Their Applications (Berkeley 1999), Math. Sci. Res. Inst. Publ. 40, Cambridge University Press, Cambridge (2001), 171–207. Search in Google Scholar
© 2016 by De Gruyter