Principle axis theorem

Ouch I was just searching the web for Principle axis theorem, and the spectral theorem.
Spectral theorem on Mathworld seems completely cryptic to me, it seems related to the linear algebra description somehow, but I’m guessing it’s the most general form of it as usual.
Not far down in the google was this page, so here’s the book I’m using: Linear Algebra - Larry Smith 3rd edition.