Eigen, singular, cosine-sine, and Autonne--Takagi vectors distributions of random matrix ensembles

TL;DR

This paper characterizes the distributions of eigen, singular, cosine-sine, and Autonne--Takagi vectors in various random matrix ensembles, revealing they are uniformly distributed on specific manifolds related to symmetry groups.

Contribution

It identifies the invariant distributions of key matrix decompositions across multiple classical random matrix ensembles, linking them to geometric manifolds.

Findings

Eigenvectors of Gaussian, Laguerre, Jacobi ensembles are uniformly distributed on the complete flag manifold.

Singular vectors of Ginibre ensembles are uniformly distributed on a product of flag and Stiefel manifolds.

Cosine-sine and Autonne--Takagi vectors are uniformly distributed on specific manifolds related to symmetry groups.

Abstract

We show that some of the best-known matrix decompositions of some of the best-known random matrix ensembles give us the unique $G$-invariant uniform distributions on some of the best-known manifolds. The eigenvectors distributions of the Gaussian, Laguerre, and Jacobi ensembles are all given by the uniform distribution on the complete flag manifold. The singular vectors distributions of Ginibre ensembles are given by the uniform distribution on a product of the complete flag manifold with a Stiefel manifold. Circular ensembles split into two types: The cosine-sine vectors distributions of circular real, unitary, and quaternionic ensembles are given by the uniform distributions on products of a (partial) flag manifold with copies of the orthogonal, unitary, or compact symplectic groups. The Autonne--Takagi vectors distributions of circular orthogonal, Lagrangian, and symplectic ensembles are given by the uniform distributions on Lagrangian Grassmannians.