If $P$ is a projection matrix such that $PX = X$ does it mean $X$ has a singular distribution?

Question

Suppose ${\bf X}$ is a $n \times 1$ vector of random variables and ${\bf P}$ is $n \times n$ a projection matrix such that ${\bf PX} = {\bf X}$. Rank(${\bf P}) < n$. So does it mean that ${\bf X}$ has a singular distribution, for example singular multivariate normal distribution?

Answer 1

Since $P$ is not full-rank, its range is a vector subspace of dimension less than n. Every vector subspace of dimension less than n has measure Lebesgue measure 0.

You say that every point in the support of the random variable $X$ is mapped by $P$ to itself. Thus, the support of the random variable $X$ is a subset of $\operatorname{ran}(P)$, which is a vector subspace of dimension less than n. Therefore, the support of $X$ has measure 0. Therefore, $X$ is a singular distribution.