Notation ${\rm {MG))_{p}(\alpha ,\beta ,{\boldsymbol {\Sigma )))$ $\alpha >{\frac {p-1}{2))$ shape parameter (real) $\beta >0$ scale parameter ${\boldsymbol {\Sigma ))$ scale (positive-definite real $p\times p$ matrix) $\mathbf {X}$ positive-definite real $p\times p$ matrix ${\frac {|{\boldsymbol {\Sigma ))|^{-\alpha )){\beta ^{p\alpha }\,\Gamma _{p}(\alpha )))|\mathbf {X} |^{\alpha -{\frac {p+1}{2))}\exp \left({\rm {tr))\left(-{\frac {1}{\beta )){\boldsymbol {\Sigma ))^{-1}\mathbf {X} \right)\right)$ $\Gamma _{p}$ is the multivariate gamma function.

In statistics, a matrix gamma distribution is a generalization of the gamma distribution to positive-definite matrices. It is a more general version of the Wishart distribution, and is used similarly, e.g. as the conjugate prior of the precision matrix of a multivariate normal distribution and matrix normal distribution. The compound distribution resulting from compounding a matrix normal with a matrix gamma prior over the precision matrix is a generalized matrix t-distribution.

This reduces to the Wishart distribution with $\beta =2,\alpha ={\frac {n}{2)).$ Notice that in this parametrization, the parameters $\beta$ and ${\boldsymbol {\Sigma ))$ are not identified; the density depends on these two parameters through the product $\beta {\boldsymbol {\Sigma ))$ .