This question has been answered before, but I want to check if my solution using minimal polynomials is good.
A projection matrix satisfies $M^2 = M$, so it satisfies the polynomial equation $M(M-1) = 0$. Thus the minimal polynomial must be either $M$, $M-1$, or $M(M-1)$. Since in all cases the polynomial factors into distinct linear factors, the matrix is diagonaliazable.
