Question: Let be a normal operator on a finite-dimensional inner product space. Prove that if is a projection, then is also an orthogonal projection.

Attempt: Since is normal, and since is a projection, then .

Since is an orthogonal projection if and only if , we already have the left equality by that fact that is a projection. We only need to show that and we'll be done.

Somehow, I figure that I can do this by showing that but how?