TITLE: Diagonalization of a projection
QUESTION [16 upvotes]: If I have a projection $T$ on a finite dimensional vector space $V$, how do I show that $T$ is diagonalizable?

REPLY [4 votes]: If $P$ is a projection and $J$ is the Jordan matrix associated with $P$ (that is, for some nonsingular $Q$, $P=QJQ^{-1}$) then 
$$J^2=\left(Q^{-1}PQ\right)\left(Q^{-1}PQ\right)=Q^{-1}P^2Q=Q^{-1}PQ=J$$
This is possible only when $J$ is diagonal.