Consider a square matrix $A$ over the real numbers that is not orthogonal. Let $v \neq 0$ be a non-zero vector such that $A^T A v = A A^T v = v$. Is it true that $v$ must be an eigenvector of $A$?
It's not hard to see that if $v$ is an eigenvector, then its eigenvalue must be on the unit circle. But I can't determine whether or not $v$ is necessarily an eigenvector.
Somehow this looks like it would be a consequence of the spectral theorem, but I can't see a crisp proof. Thanks!