Relations between generalised Wishart matrices, the Muttalib--Borodin model and matrix spherical functions

TL;DR

This paper unifies the theory of generalized Wishart matrices in real and complex cases, linking them to the Muttalib-Borodin model and matrix spherical functions, and deriving eigenvalue distributions.

Contribution

It bridges separate developments in real and complex Wishart matrices and connects them to matrix spherical functions and the Muttalib-Borodin model.

Findings

Unified theory for real and complex Wishart matrices.

Established connection to matrix spherical functions.

Derived eigenvalue probability density functions.

Abstract

Generalised uncorrelated Wishart matrices are formed out of rectangular standard Gaussian data matrices with a certain pattern of zero entries. Development of the theory in the real and complex cases has proceeded along separate line. For example, emphasis in the real case has been placed on the Bellman and Riesz distributions, while that in the complex case has been shown to be closely related to the Muttalib-Borodin model. In this work, as well as uniting the lines of development, a tie in with matrix spherical functions is identified in the context of deducing the eigenvalue probability density function from the joint element probability density function.