Integration of invariant matrices and application to statistics

TL;DR

This paper studies invariant random matrices using Weingarten calculus, deriving formulas for moments and applying them to improve understanding of pseudo-inverses of Gaussian and Wishart matrices in statistics.

Contribution

It introduces new formulas for moments of invariant matrices and applies these to derive novel results for pseudo-inverses in statistical matrix models.

Findings

Formulas for moments of invariant matrices in terms of eigenvalues

New expressions for pseudo-inverse of Gaussian matrices

Enhanced understanding of inverse Wishart matrices

Abstract

We consider random matrices that have invariance properties under the action of unitary groups (either a left-right invariance, or a conjugacy invariance), and we give formulas for moments in terms of functions of eigenvalues. Our main tool is the Weingarten calculus. As an application to statistics, we obtain new formulas for the pseudo inverse of Gaussian matrices and for the inverse of compound Wishart matrices.