# Row Space of Random Matrix

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 $$P$$?