Normal matrices and projections

Question

Just need a hint on following exercise:

Let $P \in M_n(\mathbb{C}) $ be a normal matrix such that $P^9=P^8$ . Prove $P$ is an orthogonal projection.

Thanks is advance.

Answer 1

The hint is to (unitarily) diagonalize; or to use the Spectral Theorem (both things are the same).

Orthogonal projection usually means $P^*P=P $, which is more often written as $P^2=P=P^*$.

Answer 2

The result holds in any Hilbert space. If $P$ is normal then $$\|Px\|=\|P^*x\|$$ follows from looking at squares and use normality. But then $\ker P = \ker P^*$. Now if $$P^* P x=0$$ then obviously $\|Px\|^2 = (x,P^*Px)=0$ so also $Px=0$. So $\ker P = \ker P^*P = \ker P^2$. This generalizes then to $$\ker P = \ker P^2 = ... = \ker P^8$$ So $P^9=P^8$ means $P^8(1-P)=0$ or by what has just been said $P(1-P)=0$. In other words $P^2=P$ is a projection. Since $P(1-P)=0$ also $P^*(1-P)=0$ from which we deduce that $P=P^*$ so $P$ is an orthogonal projection.