Explicit Evaluations of Matrix-variate Gamma and Beta Integrals in the Real and Complex Cases

TL;DR

This paper presents explicit evaluation methods for matrix-variate gamma and beta integrals in real and complex cases, using triangular matrix transformations to reveal their structure and facilitate calculations.

Contribution

It introduces several procedures for explicit evaluation of matrix-variate gamma and beta integrals, enhancing understanding and computation in both real and complex scenarios.

Findings

Explicit evaluation procedures for matrix-variate gamma and beta integrals.

Methods applicable to similar matrix integrals involving scalar functions.

Structural insights into matrix-variate gamma and beta integrals.

Abstract

Matrix transformations in terms of triangular matrices is the easiest method of evaluating matrix-variate gamma and beta integrals in the real and complex cases. Here we give several procedures of explicit evaluation of gamma and beta integrals in the general real and complex situations. The procedure also reveals the structure of these matrix-variate integrals. Apart from the evaluation of matrix-variate gamma and beta integrals, the procedure can also be applied to evaluate such integrals explicitly in similar situations. Various methods described here will be useful to those who are working on integrals involving real-valued scalar functions of matrix argument in general and gamma and beta integrals in particular.