Suppose that $\left\{e_{1}, \ldots, e_{3}\right\}$ is a basis of the complex vector space $\mathbb{C}^{3}$ and that $A: \mathbb{C}^{3} \rightarrow \mathbb{C}^{3}$ is the linear operator defined by $A\left(e_{1}\right)=e_{2}, A\left(e_{2}\right)=e_{3}$, and $A\left(e_{3}\right)=e_{1}$.

By considering the action of $A$ on column vectors of the form $\left(1, \xi, \xi^{2}\right)^{T}$, where $\xi^{3}=1$, or otherwise, find the diagonalization of $A$ and its characteristic polynomial.