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The Wishart distribution is a common matrix distribution on square symmetric semi-definite matrices.

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Distribution of the product of a Wishart matrix

There is a theorem which states that if $\mathbf{M} \sim W_d(\Sigma, m)$, and $m > d$, then $\frac{\mathbf{a}^T\mathbf{\Sigma}^{-1}\mathbf{a}}{\mathbf{a}^T\mathbf{M}^{-1}\mathbf{a}}$ has the $\chi_{m- …
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