michael_nielsen

A new & peculiar essay, on a (very!) unusual approach I've been using to deepen my understanding of mathematics: http://cognitivemedium.com/srs-mathematics 

My favorite way to deepen my understanding of mathematics is to repeatedly re-learn topics when I inevitably forget the details over the years. :)

Incidentally: Any operator T can be written in the form T = A + iB where A and B are self-adjoint operators. Then T* = A - iB, so T T* = A^2 - B^2 + [A, B] and T* T = A^2 - B^2 + [B, A]. So T T* = T* T iff [A, B] = [B, A] = 0 iff A and B commute.

That lets you deduce the spectral theorem for normal operators from the spectral theorem for self-adjoint operators and simultaneous diagonalization for commuting operators. That's been my preferred proof for a while, but everyone has their pet favorites for these basic theorems.