multivariate gamma function (real-valued)

The real-valued multivariate gamma function is defined by

Γm(a)=𝔖e-TrS|S|a-12(m+1)dS, (1)

where 𝔖 is the set of all m×m real, positive definitePlanetmathPlanetmath symmetric matricesMathworldPlanetmath, i.e.

𝔖={Sm×mS>0,xTSx>0xm×1{0}}. (2)

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

Γm(a)=π14m(m-1)i=1mΓ(a-12(i-1)). (3)


A. T. James, “DistributionsDlmfPlanetmathPlanetmath 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)