distribution of $\|P_VX\|^2$ with orthogonal projection $P$ onto $V$

Question

We've had the following question discussed today but without any result:

Let $X_1,\dots,X_d$ be random variables, iid and $X_n\sim N(\mu_n,1)$.

How can we describe the distribution of $\|P_VX\|^2$ with $X=(X_1,\dots,X_d)$, $V\subset \mathbb R^d$ and an orthogonal projection $P$?

Answer 1

Hint: use the fact that $$ P_V(I - P_V) = 0 $$ and the Cochran theorem.