High-Dimensional Random Matrices from the Classical Matrix Groups, and Generalized Hypergeometric Functions of Matrix Argument

Donald St P Richards

doi:10.3390/sym3030600

Abstract

Results from the theory of the generalized hypergeometric functions of matrix argument, and the related zonal polynomials, are used to develop a new approach to study the asymptotic distributions of linear functions of uniformly distributed random matrices from the classical compact matrix groups. In particular, we provide a new approach for proving the following result of D’Aristotile, Diaconis, and Newman: Let the random matrix Hn be uniformly distributed according to Haar measure on the group of n × n orthogonal matrices, and let An be a non-random n × n real matrix such that tr (A'nAn) = 1. Then, as n→∞, √n tr AnHn converges in distribution to the standard normal distribution.

Highlights

The study of high-dimensional random orthogonal and unitary matrices can be traced to a famous paper of E
Borel [1] in which the following result is proved: Let X1,n denote the first coordinate of Xn, a n-dimensional random vector that is uniformly distributed on the unit sphere S n−1 ; as n → ∞
It was striking to us that, throughout the existing literature on high-dimensional random matrices from the classical compact matrix groups, the theory of generalized hypergeometric functions of matrix argument appears not to have played an explicit role. We found this absence intriguing because it has been known since the work of Herz [15] that the characteristic function of a uniformly distributed random orthogonal matrix can be expressed in terms of the Bessel functions of matrix argument; a primary motivation for the invention of those Bessel functions was the study of random matrices which are uniformly distributed on O(n)

Summary

Introduction

The study of high-dimensional random orthogonal and unitary matrices can be traced to a famous paper of E. It was striking to us that, throughout the existing literature on high-dimensional random matrices from the classical compact matrix groups, the theory of generalized hypergeometric functions of matrix argument appears not to have played an explicit role. We found this absence intriguing because it has been known since the work of Herz [15] that the characteristic function of a uniformly distributed random orthogonal matrix can be expressed in terms of the Bessel functions of matrix argument; a primary motivation for the invention of those Bessel functions was the study of random matrices which are uniformly distributed on O(n). By application of a result of Johansson [6], we will obtain an upper bound on the distance, in the supremum norm on R, between a certain generalized hypergeometric function of scalar matrix argument and the Gaussian quantity, exp(−t2 /2), t ∈ R

Zonal Polynomials and a Generalized Hypergeometric Function of Matrix Argument

The Case of the Stiefel Manifold

The Case of the Orthogonal Group

The Case of the Unitary Group