Let $V$ be a complex vector space with basis $\left\{e_{1}, \ldots, e_{n}\right\}$. Define $T: V \rightarrow V$ by $T\left(e_{i}\right)=e_{i}-e_{i+1}$ for $i<n$ and $T\left(e_{n}\right)=e_{n}-e_{1}$. Show that $T$ is diagonalizable and find its eigenvalues. [You may use any theorems you wish, as long as you state them clearly.]