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multivariate gamma function (complex-valued)

(Definition)

The complex multivariate gamma function is defined as

$\displaystyle \tilde{\Gamma}_m(a)=\int_{\mathfrak{A}} e^{-{\mathrm{Tr}}A}\vert A\vert^{a-m} {\rm d}A,$

(1)

where $\mathfrak{A}$ is the set of all $m \times m$ positive, complex-valued Hermitian matrices, i.e.

$\displaystyle \mathfrak{A} = \left\{A \in \Bbb{C}^{m \times m} \vert A = A^H, A > 0\right\}.$

(2)

It can also be expressed in terms of the gamma function as follows

$\displaystyle \tilde{\Gamma}_m(a)=\pi^{{1 \over 2}m(m-1)} \prod\limits_{i=1}^{m}\Gamma(a-i+1).$

(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.

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Also defines:

gamma function (multivariate complex)

Keywords:

Gamma multivariate complex

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Cross-references: terms, Hermitian matrices, positive, gamma function, complex

This is version 11 of multivariate gamma function (complex-valued), born on 2004-05-13, modified 2006-10-17.
Object id is 5854, canonical name is MultivariateGammaFunctionComplexValued.
Accessed 2763 times total.

Classification:

62H10 (Statistics :: Multivariate analysis :: Distribution of statistics)

Pending Errata and Addenda

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