Prove/Disprove: $v$ is an eigenvector of $T^n$ implies $v$ is an eigenvector of $T$

Question

Prove/Disprove: $v$ is an eigenvector of $T^n$ implies $v$ is an eigenvector of $T$

I'm pretty sure it's not necessarily true, but can't think of a counter example.

Can you help me think of one?

Answer 1

The matrix $$ T=\pmatrix{0&1\\0&0} $$ has a one-dimensional eigenvector space spanned by $[1,0]^T$, but any 2-vector is an eigenvector of $T^2=0$.

Answer 2

Hint Take $T$ the rotation of angle $\frac{2\pi}n$.