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Is there an established algorithm for sampling from the Matrix-Gamma distribution (https://en.wikipedia.org/wiki/Matrix_gamma_distribution)? I suspect the procedure would be similar to generating from a Wishart distribution (i.e. https://www.math.wustl.edu/~sawyer/hmhandouts/Wishart.pdf), except using gamma distributed random variables rather than $\chi^2$ random variables on the diagonal. I cannot find a reference, however.

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The matrix Gamma distribution (as defined in Wikipedia) $MG_p(\alpha,\beta,\Sigma)$ is nothing but the Wishart distribution $W_p\left(2\alpha, \frac{\beta}{2}\Sigma\right)$.

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