Let $V$ be a finite-dimensional vector space over $C$, let $T:V\rightarrow V$ be a linear operator, and let $M$ be a nontrivial subspace of $V$ that is invariant under $T$. Prove that $M$ contains an eigenvector of T.
I am trying to find the eigenvector, but I can not.
Hint: As $M$ is $T$-invariant, we may consider $T|_M \colon M \to M$, now $T|_M$ is an endomorphism of a non-trivial $\mathbb C$-vector space. What can you say about eigenvectors of $T|_M$?