Let $V$ be a finite-dimensional inner product space, and let $E$ be an idempotent linear operator on $V$, i.e., $E^2 = E$. Prove that E is self-adjoint if and only if $EE^* = E^*E$.
Are there any simpler answers to the question that the answers provided here Normal, idempotent operator implies self-adjointness. . Both answers seem to be correct but contain logical steps that I can't comprehend e.g $(I−E)Ex=0 \Rightarrow (I−E^∗)Ex=0 $ and $v^\ast E^\ast Ev=0 \Rightarrow Ev=0$