A Mathematician’s Apology - G.H. Hardy
I just reread the last half of this book, and it’s clarified things about maths for me. I think it made a big difference reading this now that I have a lot more understanding of mathematics.
The Principle axis theorem.
First.. The spectral theorem, the general proof presented is by induction.
Proof of the 2×2 and 3×3 case. Then for the NxN case the hard part is to show that an eigenvector exists. Once that’s done you normalise it, throw a bunch of orthonormal vectors onto it as a basis, and the new matrix will look like:
e 0 ... 0 0 . B 0
Where e is an eigenvector and B is a symmetric matrix, then by induction bla bla..
Spectral theorem is proved.