If $P$ is an orthogonal projection, prove that $P^+ = P$. Where '$^+$' indicates the Moore-Penrose pseudo-inverse.
What we know is the $P$ is symmetric and idempotent. That is $P = P^2 = P^T$. I am assuming we just need to check the Penrose conditions?
- $AGA = A \implies PPP = P^2P = PP = P^2 = P$
- $GAG = G \implies PPP = P$ (as above :)).
- $(AG)^T = AG \implies (PP)^T = (P^2)^T = (P)^T = P^T = P = P^2 = PP$
- $(GA)^T = GA\implies (PP)^T = PP$ (as above :)).
Is this what the proof intended? Please let me know if some different was desired.