Overview

Every result in Stage 3 so far required a square matrix — $A\mathbf{x}=\lambda\mathbf{x}$ makes no sense unless $A\mathbf{x}$ and $\mathbf{x}$ live in the same space. But most matrices built from real data are not square: a returns matrix has many more trading days (rows) than assets (columns); a factor-loading matrix has more instruments than factors. This closing topic of the unit previews the decomposition that removes the square restriction entirely — the singular value decomposition $A=U\Sigma V^\top$, which exists for every matrix, square or not, defective or not. It is deliberately a preview only: the full construction, the proof of existence, and the deep consequences (low-rank approximation, the pseudoinverse, numerical stability) are the central subject of learn/library/mathematics/linear-algebra-ii/.