Let $E$ be a $\mathbb{C}$-vector space of dimension $n \geq 1$. We say that an endomorphism $u$ of $E$ is a permutation endomorphism if there exists a basis $(e_1, \ldots, e_n)$ of $E$ and a permutation $\sigma \in \mathfrak{S}_n$ such that $u(e_j) = e_{\sigma(j)}$ for all $j \in \llbracket 1, n \rrbracket$.
Show that $u$ is a permutation endomorphism if and only if there exists a basis in which its matrix is a permutation matrix.