Eigenmumbles

It’s the start of a new academic year. Among other things, that means that I get to introduce myself to new students and to tell them briefly about the things I work on. Brevity and humor go a long way in such introductions, so I usually just say that I’m obsessed with eigenvalues, and give one or two examples of projects I work on that boil down to spectral analysis – preferably the types of problems where the connection isn’t completely obvious, since those tend to draw more interest and lead to more interesting follow-up conversations.

Of course, just because I’m obsessed with eigenvalues doesn’t mean anyone else should be. Furthermore, if I really was only interested in these things out of some psychological abnormality, I probably wouldn’t be particularly good at interesting anyone else, and getting other people interested is a big part of the game. So, other than obsession, why do I think eigenvalues are interesting? My top three reasons are:

1. Eigenvalue analysis lets me decouple the dynamics of systems. Instead of having to understand the evolving interactions of a hundred different quantities, I can focus on one interaction mode at a time. This sort of analysis is tremendously powerful even if I’m interested in nonlinear dynamical systems, since I can use eigenvalues and eigenvectors to describe the start of phenomena like bucking or the birth of unstable oscillations.
2. Eigenvalue problems are a type of nonlinear equation that I know how to solve. So if I can turn your problem into an eigenvalue problem, I can probably figure out how to solve it. Because there is an intimate connection between eigenvalues and polynomials, I can turn an awful lot of problems involving polynomials into eigenvalue problems, so it’s not so implausible that I might be able to turn your problem into an eigenvalue problem.
3. Similarly, symmetric eigenvalue problems are almost the only non-convex optimization problem where I know how to find globally optimal solutions. Consequently, when I find things that aren’t actually eigenvalue problems but are something much harder, I’m tempted to take off my glasses and ask if it doesn’t look like an eigenvalue problem after all. Enough other people have this same temptation that it has a special name: continuous relaxation.