Orthogonal projection matrix eigenvector

Question

Let $P$ be $n\times n$ orthogonal projection matrix. Suppose $x$ is an $n$ dimensional unit vector which belongs to the range space of $P$. Now, I want to show the following holds:$$Px=x.$$This equality says that one of the eigenvalues is $1$. Showing this easy because if we let eigenvector of this system $v$ then:$$Pv=\lambda v\implies PPv=\lambda Pv\implies Pv=\lambda ^2v\implies \lambda v=\lambda ^2v,$$which yields $\lambda _{1,2}=0,1$, because for an orthogonal projection matrix we know that $P^2=P$. On the other hand, I'm not sure how to show that $x$ is indeed an eigenvector.

Answer 1

Any - not necessarily orthogonal - projection (i.e. idempotent) matrix satisfies this:

If $x$ is in the range of $P$ then $x=Py$ for some $y$, thus $$x\ =\ Py\ =\ P^2y\ =\ Px\,.$$