Let $E$ be a $\mathbb{C}$-vector space of dimension $n \geq 1$. 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.
Let $E$ be a $\mathbb{C}$-vector space of dimension $n \geq 1$. 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.