I'm trying to solve this really interesting problem, taken from "Berkeley Problmes in Mathematics" by Souza and Silva
Trying to make headway, I've noticed that if the last two statements are correct, then $M^T(Mu)=\sigma^2 u$, so in other words, there exists an eigenvector $u$ with the eigenvalue $\sigma^2$. Therefore, I have two questions, and I would dearly love a proof or counter-example:
- Is it true that $M^TM$ must have at least a (real) eigenvector?
- I strongly suspect this $\sigma$ they talk of is the operator norm of $M$. Is this true?
- Is this a dead-end way to try to solve this problem. Should I try something else?
Of course, everything is real-valued here.