Let $\Sigma$ be an $p\times p$ dimensional covariance matrix that is distributed Inverse Wishart with degrees of freedom $\nu$ and Prior scale matrix $\Psi$ such that we write $\Sigma \sim W^{-1}(\nu, \Psi)$. Also let $A$ be a $q \times p$ matrix. What is the distribution of $A \Sigma A^T$. Please use the parameterization of the Inverse Wishart from Wikipedia in your answer.

Is the answer just $A \Sigma A^T \sim W^{-1}(\nu, A\Psi A^T)$? For some reason, I am having trouble finding any references with this or any result for the Inverse Wishart (I see plenty for the Wishart).

In case it matters, what about for the special case where $q=p$ and $A$ is invertible?


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