Singular Values and Evenness Symmetry in Random Matrix Theory

TL;DR

This paper explores the decomposition of singular value distributions in complex Hermitian and real symmetric random matrices with specific symmetries and weight functions, revealing new inter-relations and decompositions across matrix ensembles.

Contribution

It provides new decompositions of singular value distributions for real symmetric matrices with orthogonal symmetry and specific weights, extending known results to additional matrix ensembles.

Findings

Decomposition of singular values for real symmetric matrices with orthogonal symmetry.

Inter-relations between gap probabilities for orthogonal and unitary symmetries.

Decomposition results for circular ensembles via stereographic projection.

Abstract

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.