Eigenvalues, but Make It Cute

The one direction that survives

An eigenvector is a direction a matrix leaves alone, up to scaling:

$A\mathbf{v} = \lambda \mathbf{v}$

The scale factor $\lambda$ is the eigenvalue. Everything else about $A$ is bookkeeping around these special directions.

Where they hide

PCA, PageRank, and the stability of every dynamical system you’ll ever meet are all eigenvalue questions wearing different hats.