# multivariate gamma function (real-valued)

The real-valued multivariate gamma function is defined by

$${\mathrm{\Gamma}}_{m}(a)={\int}_{\U0001d516}{e}^{-\mathrm{Tr}S}{\left|S\right|}^{a-\frac{1}{2}(m+1)}dS,$$ | (1) |

where $\U0001d516$ is the set of all $m\times m$ real, positive definite^{} symmetric matrices^{}, i.e.

$$\U0001d516=\{S\in {\mathbb{R}}^{m\times m}\mid S>0,{x}^{\mathrm{T}}Sx>0\forall x\in {\mathbb{R}}^{m\times 1}\setminus \{0\}\}.$$ | (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) |