How to remember the fact that $P^2=P$ and $P^*=P\iff P$ is an orthogonal projection.

Question

In functional analysis,I have studied the fact that for an operator $P:H\to H$ where $H$ is a complex Hilbert space, $P^2=P$(Idempotent) and $P^*=P$(Self-adjoint)$\iff P$ is an orthogonal projection.But I cannot find the motivation to remember that $P$ has to be self-adjoint (For idempotent the motivation is that $P$ acts like identity on its image).So,is there some visual interpretation of this property?If yes,then can someone please state that.

Answer 1

Orthogonal projections are characterized within general projections through the important property $Px \perp x-Px$ for all $x \in H$. This property follows directly from $P^* = P$ since \begin{align} \langle Px,x-Px\rangle &= \langle Px,x\rangle - \langle Px,Px \rangle \\ &= \langle Px,x\rangle - \langle P^*Px,x \rangle \\ &= \langle Px,x\rangle - \langle P^2x,x \rangle\\ &= \langle Px,x\rangle - \langle Px,x \rangle\\ &= 0. \end{align}