Is every eigenvector of AA an eigenvector of A? Let $V$ be a (finite-dimensional) vector space and $A \colon V \to V$ a linear map. 
Is it true that, if $v$ is an eigenvector of $A\circ A$, then $v$ is an eigenvector of $A$?
I know the converse statement is true.

Hint: Consider
$$A=\begin{pmatrix}1&0\\0&-1\end{pmatrix}. $$
Or
$$A=\begin{pmatrix}0&1\\0&0\end{pmatrix}. $$
