TITLE: Is a projector matrix the inverse of itself?
QUESTION [10 upvotes]: I want to confirm if a projector matrix is its own inverse. I have $x=Px$ and $Px=P^2x$, so premultiplying the second equation with $P^{-1}$ twice, I get $P^{-1}x=Px$ for all x, implying $P^{-1}=P$. Is this reasoning correct?
So are all projection matrices orthogonal too?

REPLY [2 votes]: If by "projector matrix" you mean the matrix of a projection onto a (proper) subspace, then the rank of such a matrix will be the dimension of that subspace, which is less than the number of columns. So the nullspace is nontrivial, and the matrix is not invertible.