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.