Suppose that $T:V\to V$ is a linear transformation and $V$ is a vector space over $\mathbb{C}$. $W$ is a non-zero subspace of $V$ such that $T(W)\subset W$, that is $W$ is T-invariant. Does it mean that $T$ has a non-zero eigen vector in $W$?
I am not at all sure about where the answer is in positive. I also cannot find any counter. If I take for example $W=V$, then the hypothesis always hold but there exists non-zero eigen vector always due to the field being $\mathbb{C}$. But, I don't know what to do in case of arbitrary invariant subspace.
Thanks in advance for any help.