Proposition 3

Let $E$ be a vector space of finite dimension $n$ over a commutative field $K$, and let $u$ be an endomorphism of $E$; for every monic irreducible polynomial $p(X)$, let $M_p$ be the vector subspace consisting of elements $x$ of $E$ such that $(p(u))^k \cdot x = 0$ for some integer $k$. Then $M_p$ is closed under $u$, the vector space $E$ is the direct sum of the $M_p$, and there exist polynomials $s_p$ such that, for all $x \in E$, the component of $x$ in $M_p$ is equal to $s_p(u) \cdot x$.