How to generate random matrices from the classical compact groups

TL;DR

This paper presents simple algorithms for generating random matrices from classical compact groups like U(N), O(N), and USp(N), useful for applications in random matrix theory and related fields.

Contribution

It introduces straightforward methods to generate random matrices from classical groups using standard linear algebra, extending to Dyson circular ensembles.

Findings

Algorithms are easy to implement with standard linear algebra packages.

Methods apply to Dyson circular ensembles.

Provides accessible exposition for a general mathematical audience.

Abstract

We discuss how to generate random unitary matrices from the classical compact groups U(N), O(N) and USp(N) with probability distributions given by the respective invariant measures. The algorithm is straightforward to implement using standard linear algebra packages. This approach extends to the Dyson circular ensembles too. This article is based on a lecture given by the author at the summer school on Number Theory and Random Matrix Theory held at the University of Rochester in June 2006. The exposition is addressed to a general mathematical audience.