Let $\displaystyle P \in \textbf{C}^{m\times m}$ be a projector. We prove that $\displaystyle \left\| P \right\| _{2}\geq 1$ , with equality if and only if P is an orthogonal projector.

I suppose we could use the formula $\displaystyle \left\| P \right\| _{2}= max_{\left\| x \right\| _ {2} =1} \left\| Px \right\| _{2}$

and use the fact that $\displaystyle P^{2}=P$

But I am not sure how to proceed.