Proposition 4

— Let E be a vector space of finite dimension n over a commutative field K, and let u be an endomorphism of E. Then there exist n monic polynomials $q_i(X) \in K[X]$ ($1 \leq i \leq n$) such that $q_i$ divides $q_{i+1}$ for $1 \leq i \leq n-1$, and E is the direct sum of n subspaces $V_i$ ($1 \leq i \leq n$) closed under u, cyclic (with respect to u), and such that the minimal polynomial of the restriction of u to $V_i$ is equal to $q_i$ ($1 \leq i \leq n$). The polynomials $q_i$ are uniquely determined by these conditions, and $q_n$ is the minimal polynomial q of u.