# Is a projector matrix the inverse of itself?

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?

• Not all projections are invertible. Nov 4, 2012 at 22:21
• But $tr(P)=rank(P)=k$ for a $k\times k$ projection matrix, right? Nov 4, 2012 at 22:23
• Consider the $2\times 2$ projection matrix $$P=\left(\begin{array}{cc}1 & 0\\0 & 0\end{array}\right)$$ for an easy example of a non-invertible projection matrix. Nov 4, 2012 at 22:27
• @Shyam It's true that for projections $\rm{tr}(P) = \rm{rank}(P)$ but it's not true that the matrix is always full rank.
– EuYu
Nov 4, 2012 at 22:28
• Thanks all of you. Yes, I have $P=X(X^TX)^{-1}X^T$, where $X$ is full-rank. The counterexamples are illuminating. Nov 4, 2012 at 22:29

Projector matrices are idempotent, and as a rule, need not have an inverse at all (since it will usually have a non-trivial nullspace). For $P$ to be its own inverse, we need $P^2=I$. Since $P^2=P$ for any projector matrix, then the only projector matrix that is its own inverse is the identity (which we can think of as the trivial projector of a space onto itself).