05 · Invariants

Eigen

Some arrows only stretch. Those are eigenvectors; the stretch is the eigenvalue. A rotation in the plane has none that are real — nothing to grab.

In English

Some arrows go into a machine and come out only longer or shorter — they do not turn. Those are eigenvectors. The stretch factor is the eigenvalue. A pure rotation in 2D has no real ones: every arrow turns.

Try this: Pick a stretchy map. Drag until the arrow stops turning. Then pick a rotation and notice you never find one.
Keep this: Eigen = “this arrow only stretches.” Not every map has one you can see.

Lab · arrows that only stretch

[[2, 0], [0, 0.5]]

λ = 2, 0.5

Solve Av = λv. If v turns, it is not an eigenvector. A 90° rotation has no real v that survives — the characteristic polynomial has no real root.

PCA, PageRank, Grover’s 2D rotation, and “the model’s residual stream” are all eigen-stories. Once you can see one, you can see the others.