In this subsection, $E$ is 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 $j \in \llbracket 1, n \rrbracket$. Let $u$ be an endomorphism of $E$ such that $u^2 = \operatorname{Id}_E$. Show that $u$ is a permutation endomorphism if and only if $\operatorname{Tr}(u)$ is a natural integer.
In this subsection, $E$ is 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 $j \in \llbracket 1, n \rrbracket$.
Let $u$ be an endomorphism of $E$ such that $u^2 = \operatorname{Id}_E$. Show that $u$ is a permutation endomorphism if and only if $\operatorname{Tr}(u)$ is a natural integer.