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: If $$E$$ is idempotent ($$E^2 = E$$) and normal ($$E^{\ast} E = E E^{\ast}$$), then $$E$$ is self-adjoint…

I am currently taking a course on communication complexity with Alexander Sherstov. Much of communication complexity involves matrix analysis, so yesterday we did a brief review of results from linear algebra. In the review, Sherstov gave the following definition for the rank of a matrix $$M \in \mathbf{F}^{n \times m}$$: $\mathrm{rank}(M) = \min{k | M = A B, A \in \mathbf{F}^{n \times k}, B \in \mathbf{F}^{k \times m}}$ Since this “factor” definition of rank appears very different from the standard definition given in a…