Proposition 6

— Let $E$ be a vector space of finite dimension $n$ over a commutative field $K$, and let $u$ be an endomorphism of $E$. Then $E$ is the direct sum of subspaces $E_k$, closed under $u$ and cyclic with respect to $u$, such that the minimal polynomial of the restriction of $u$ to $E_k$ has the form $p_k^{n(k)}$, where $p_k$ is an irreducible polynomial, and $E_k$ cannot be expressed as a direct sum of two nonzero subspaces each closed under $u$. For every monic irreducible polynomial $p \in K[X]$ and every integer $n \geq 1$, the number $m(p^n)$ of subspaces $E_k$ in any such decomposition, such that $p^n$ is the minimal polynomial of the restriction of $u$ to $E_k$, is uniquely determined.

The $p_k^{n(k)}$ determine the similarity invariants of $u$ and vice versa; we can get from one to the other by the procedure explained in VII, p. 25, Remarks 2 and 3. Furthermore, we can immediately get from the decomposition considered in Prop. 6 to those considered in Prop. 3 and 4.

Note that the monic irreducible polynomials $p \in K[X]$ such that $m(p^n) > 0$ for some integer $n \geq 1$ are precisely the monic irreducible factors of the minimal polynomial of $u$. Thus, in contrast to the similarity invariants, these polynomials depend in general on the field $K$ in which we are working.