Two Tricky Linear Algebra Problems

There are a couple questions from the homework that seem to have given people (myself included) a fair amount of trouble. Since I wasn’t able to give satisfactory answers to these questions in office hours, I thought I’d write up clean solutions to the problems, available here. The questions both involve projections \(E : V \to V\) where \(V\) is an inner product space. The problems ask you to prove:

  1. If \(E\) is idempotent (\(E^2 = E\)) and normal (\(E^* E = E E^*\)), then \(E\) is self-adjoint (\(E = E^*\)).
  2. If \(E\) satisfies \(\|E v\| \leq \|v\|\) for all \(v \in V\), then \(E\) is an orthogonal projection onto its image.