When a matrix transforms the plane, most vectors get knocked off their original line — they rotate as well as stretch. But a few special directions are different: the transformation only stretches them, leaving them pointing exactly the same way (or exactly opposite). Those rare, unmoved directions are the eigenvectors of the matrix.

For an eigenvector \vec{v}, applying A is the same as multiplying by a single number \lambda — its eigenvalue, the stretch factor:

Hunt for the unmoved directions

Spin the input vector \vec{v} (faint) and watch its image A\vec{v} (bold). Most of the time the two point different ways. But at a couple of special angles they snap onto the same line — there \vec{v} is an eigenvector, and the readout shows its eigenvalue \lambda (how many times longer A\vec{v} is).

Why anyone cares

Eigenvectors are the "natural axes" of a transformation — the directions along which it does nothing but scale. Find them and a tangled matrix becomes simple: it's just stretching along its own private set of axes. That insight powers diagonalization, the stability of physical systems, the ranking behind Google's PageRank, and the principal directions of a dataset. First, though, we need a way to compute them.