## 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.