On the Distribution of an Arbitrary Subset of the Eigenvalues for some Finite Dimensional Random Matrices

TL;DR

This paper derives new compact formulas for the joint distribution and moments of arbitrary subsets of eigenvalues in finite-dimensional complex Wishart, double Wishart, and Gaussian Hermitian random matrices, enhancing understanding of their spectral properties.

Contribution

It introduces novel expressions for the joint distribution and expectations of eigenvalues using a tensor pseudo-determinant operator, applicable to both ordered and unordered cases.

Findings

Derived compact joint distribution formulas for eigenvalues.

Established expectation formulas for functions of eigenvalues.

Applicable to complex Wishart, double Wishart, and Gaussian Hermitian matrices.

Abstract

We present some new results on the joint distribution of an arbitrary subset of the ordered eigenvalues of complex Wishart, double Wishart, and Gaussian hermitian random matrices of finite dimensions, using a tensor pseudo-determinant operator. Specifically, we derive compact expressions for the joint probability distribution function of the eigenvalues and the expectation of functions of the eigenvalues, including joint moments, for the case of both ordered and unordered eigenvalues.