Showing posts tagged linear algebra

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…

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Factor Characterization of Matrix Rank

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…

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