Sampling the eigenvalues of random orthogonal and unitary matrices

TL;DR

This paper presents an efficient algorithm for directly sampling eigenvalues of Haar-distributed orthogonal and unitary matrices, enabling faster and adaptable eigenvalue sampling for various matrix groups.

Contribution

It introduces a novel factorization-based method for eigenvalue sampling that is computationally efficient and adaptable to different matrix groups.

Findings

Quadratic complexity in matrix size for eigenvalue sampling

Applicable to special orthogonal and unitary groups

Can sample matrices with specified determinant constraints

Abstract

We develop an efficient algorithm for sampling the eigenvalues of random matrices distributed according to the Haar measure over the orthogonal or unitary group. Our technique samples directly a factorization of the Hessenberg form of such matrices, and then computes their eigenvalues with a tailored core-chasing algorithm. This approach requires a number of floating-point operations that is quadratic in the order of the matrix being sampled, and can be adapted to other matrix groups. In particular, we explain how it can be used to sample the Haar measure over the special orthogonal and unitary groups and the conditional probability distribution obtained by requiring the determinant of the sampled matrix be a given complex number on the complex unit circle.