Prove projection is self adjoint if and only if kernel and image are orthogonal complements. Let V be an IPS and suppose π:V→V is a projection so that V=U⊕W (ie V=U+W and U∩W={0}) where U=ker(π) and W=im(π), and if v=u+w (with u∈U, w∈W) then π(v)=w.

Is orthogonal projection self adjoint?

(i) P is self-adjoint (ii) P is normal (iii) x − Px is orthogonal to Px for every x ∈ H. If these conditions hold then P is the orthogonal projection onto its image. Proof. If P is self-adjoint then of course P is normal.

Are projection matrices orthogonal?

(b) Every projection matrix is an orthogonal matrix.