Matrix gamma distribution

In statistics, a matrix gamma distribution is a generalization of the gamma distribution to positive-definite matrices. It is effectively a different parametrization 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.
A matrix gamma distributions is identical to a Wishart distribution with
Notice that the parameters and are not identified; the density depends on these two parameters through the product.