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Say we have a Gaussian distribution in $\mathbb{R}^p$, $x \sim N(0, \Sigma)$, and a set of $n < p$ samples from this distribution, $X \in \mathbb{R}^{n \times p}$. The projector onto the row-space of $X$ is given by $P = X^T (X X^T)^+ X$. The distribution of $(X^T X)^+$ was found by Diaz-Garcia et al. Is there an explicit form for the distribution of $P$?

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