Overview

Every topic so far has treated a matrix as a machine that scrambles vectors — $A\mathbf{x}$ mixes components, changes direction, tangles inputs together. This topic asks the question the unit's conceptual spine promised from the start: are there directions a linear map does not scramble — directions it merely stretches or shrinks? Those directions are eigenvectors, the associated stretch factors are eigenvalues, and finding them turns out to be the single most powerful move available for understanding what a matrix does, repeated across probability (Markov chains), differential equations, and — previewed at the topic's end — every method for extracting statistical structure from data.