2-norm of a projector is greater than 1

[math8]

Let P \in \textbf{C}^{m\times m} be a projector. We prove that \left\| P \right\| _{2}\geq 1 , with equality if and only if P is an orthogonal projector.I suppose we could use the formula \left\| P \right\| _{2}= max_{\left\| x \right\| _ {2} =1} \left\| Px \right\| _{2} and use the fact that P^{2}=P and P=P^{*} (P* is the transpose conjugate of P).

But I am not sure how to use these.

 

[jbunniii]

Technically this isn't true since P = 0 is a projection matrix with ||P||_2 = 0. But assuming P \neq 0, there exists some v \neq 0 such that

Pv = v

If you use this fact along with the definition of the 2-norm of a matrix, your inequality follows pretty much immediately.