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&#x27;s this amazingly non-obvious but remarkably powerful structure hidden inside matrices that I&#x27;d never even been aware of before.<p>Quantum mechanics is a huge application of eigenvectors, but I also really enjoy things like the &quot;moment of inertia tensor&quot;, whose eigenvectors are the natural axes of rotation. Or better still, coupled oscillators, where the eigenvectors give &quot;normal modes&quot; of vibration. (And if you look at coupled first-order differential equations, eigenvectors can tell you all sorts of things about &quot;trajectories&quot; of the solutions. There are great applications of that to things like population dynamics in biology.)