Definition 8

Let $E$ be a finite dimensional vector space over a commutative field and let $u$ be an endomorphism of $E$. Then a Jordan decomposition of $u$ is a pair $(u_s, u_n)$, where $u_s$ is a semi-simple endomorphism of $E$ and $u_n$ is a nilpotent endomorphism of $E$, such that $u_s u_n = u_n u_s$ and $u = u_s + u_n$.