
[Aug. 4th, 200709:26 pm]
Jason

Mostly shitty headachey sort of day, but had a nice diversion from it hanging out with Matus for a little bit, during which he told me about his adventures with (and reasons to be somewhat skeptical of) spectral clustering methods, and he lent me his copy of the Mallat book on signal processing and wavelets, which is kind of blowing my mind. I basically spent the rest of the day trying to believe the very first sentence on page 2, which, to paraphrase, says that the reason the Fourier transform is so canonical is that eigenvectors of timeinvariant linear operators on signals are precisely the sine waves e^{iwt}. I recommend the exercise of working this out for the finite case, in which case the statement is that: for any linear operator R^{n} → R^{n} that commutes with "rotations" (picture vectors in R^{n} as n real numbers arranged in a circle, not in a column) it has an orthonormal eigenbasis whose kth vector has jth component e^{2πijk/n}. Hint: (1) first show that any eigenvector of the rotation operator itself must be an eigenvector of any rotationinvariant operator as well, albeit with a different eigenvalue. (2) Show that the sine waves are eigenvectors of rotation (3) Remember a few facts from kindergartenlevel complex analysis (this part actually took me embarrassingly long) to see that they're all orthonormal. 

