The Householder-John Theorem

Let and are real matrices such that
both and are symmetric positive definite.
Define .
Then

Proof

Let be an eigenvalue of
and a (possibly complex) eigenvector.
Write:

As we find that so:

By taking the we also find:

Note that for a symmetric positive definite matrix :

Where:

So if we write for and real:

Apply this to :

As and we find