Suppose $\Psi$ is an invertible $n\times n$ matrix that does not have only real eigenvalues. I read that such a matrix can be written as a finite composition of elementary matrices with real eigenvalues. Specifically:
Every automorphism of $\mathbb{R}^n$ is a finite composition of automorphisms with real eigenvalues (elementary matrices).
I have not been able to find a proof of this, does anyone know how to show it?
It's a standard result from introductory linear algebra that, by multiplying by elementary matrices, you can row reduce any matrix into row reduced echelon form.
In the case of an invertible matrix, the row reduced echelon form is the identity matrix, which is also an elementary matrix, and so you get the theorem.