multivariate gamma function (real-valued)

The real-valued multivariate gamma function is defined by

$${\mathrm{\Gamma }}_m(a)={\int }_{𝔖}{e}^{-\mathrm{Tr}S}{\left|S\right|}^{a-\frac{1}{2}(m+1)}dS,$$ (1)

where $𝔖$ is the set of all $m\times m$ real, positive definite symmetric matrices, i.e.

$$𝔖=\{S\in {ℝ}^{m\times m}\mid S>0,x^{\mathrm{T}}Sx>0\forall x\in {ℝ}^{m\times 1}\setminus \{0\}\}.$$ (2)

The real-valued multivariate gamma function can also be expressed in terms of the gamma function as follows

$${\mathrm{\Gamma }}_m(a)={\pi }^{\frac{1}{4}m(m-1)}\prod _{i=1}^m\mathrm{\Gamma }\left(a-\frac{1}{2}(i-1)\right).$$ (3)

Reference

A. T. James, “Distributions of matrix variates and latent roots derived from normal samples,” Ann. Math. Statist., vol. 35, pp. 475-501, 1964.

Title	multivariate gamma function (real-valued)
Canonical name	MultivariateGammaFunctionrealvalued
Date of creation	2013-03-22 14:22:06
Last modified on	2013-03-22 14:22:06
Owner	rspuzio (6075)
Last modified by	rspuzio (6075)
Numerical id	15
Author	rspuzio (6075)
Entry type	Definition
Classification	msc 62H10
Defines	gamma function (multivariate real)