Attack of the Eigenthings

that would have made a great 1950s sci fi movie, I think.

Well, Nate, may the Eigenforce be with you.

Thanks!

Our professor calls it

*eigenhunting*. You could fill a dictionary with the number of words accepted by mathematicians that have "eigen" as a prefix. Off of the top of my head -- eigenvalue, eigenvector, eigenspace, eigencondition, eigenproblem, eigenfunction, eigenface (yes, that is really one related to image processing), eigensolution, eigendecomposition, eigenmode, eigenstate, eigenpair, eigenbasis ... eigen

*etc*...

It's a problem that has fascinated me in math since I first learned it. Since I have a bit of a backwards education when it comes to math, I first learned the eigenvalue problem in the context of solving systems of linear ODEs. I didn't think much of it until we started learning phase plane analysis of systems. Then I was intrigued. Why does a system evolve along the direction of its eigenvectors in the phase plane? What is the meaning of this? Then, when I took a class on nonlinear dynamics (still no linear algebra yet!) it made more sense.

*Then* I took classes in linear algebra, system dynamics and controls, and even vector spaces (the latter was way over my head as a non-math major). It all made sense (I'm a very slow learner, so it took many classes for me to "get it"). I must have seen the stuff in 10 different courses by now and it still fascinates me.

What's more is we covered the Sturm-Liouville problem last year (we'll do it again this year since I'm re-taking the class for various reasons). Now we have

*function* spaces? Until then, I had only heard of them!

It's all based off of such a simple problem that could be explained in words to someone even with no mathematical background. Even with the math, the basic problem is a simple computation that a high school student could do (though, they probably wouldn't know that they could). But, it leads to so many things that are just -- well -- beautiful.

An interesting (though not-mathematically-rigorous) visual example:

EDIT: I think it went well. Last year I got an 87/100 which wasn't bad considering the class average was a 57/100. I was irritated with myself for not knowing a proof that the professor specified in the syllabus. I'm hoping for above a 95 this year (my standards are not usually this high, but since it isn't unreasonable since I've taken the class before). I forgot this one thing about quadratic form, but it was very minor. I might have still done it correctly.