April 2005

Principle axis theorem.
Every symmetric matrix is diagonalizable.

In the more general form, for every self adjoint linear transformation, a basis can be found for which the transformations matrix representation is a diagonal matrix (ie only entries on the diagonal).

Prove it, or outline the proof.
Give an example. How do we diagonalize a matrix?
What are the implications of this theorem?

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.