[Steuard]

I love eigenvectors. For me, the rest of linear felt like a bunch of rote rules and algorithms for figuring out vaguely-interesting stuff. And then suddenly I find out that there's this amazingly non-obvious but remarkably powerful structure hidden inside matrices that I'd never even been aware of before.

Quantum mechanics is a huge application of eigenvectors, but I also really enjoy things like the "moment of inertia tensor", whose eigenvectors are the natural axes of rotation. Or better still, coupled oscillators, where the eigenvectors give "normal modes" of vibration. (And if you look at coupled first-order differential equations, eigenvectors can tell you all sorts of things about "trajectories" of the solutions. There are great applications of that to things like population dynamics in biology.)