Proposition 7

— Let $E$ be a finite dimensional vector space over a commutative field $K$, and let $u$ be an endomorphism of $E$ whose eigenvalues all belong to $K$. For each eigenvalue $a$ of $u$, let $M_a$ be the vector subspace of $E$ consisting of those elements $x$ for which there exists an integer $k \geq 1$ such that $(u - a)^k \cdot x = 0$. Then $M_a$ is closed under $u$, the vector space $E$ is the sum of the $M_a$, and there exist polynomials $s_a \in K[X]$ such that, for all $x \in E$, the component of $x$ in $M_a$ is equal to $s_a(u) \cdot x$.

The submodule $M_a$, being finitely generated as a $K[X]$-module, then has an annihilator of the form $(X - a)'$; in other words, there exists an integer $r \geq 1$ such that

for all $x \in M_a$; the restriction of $u - a$ to $M_a$ is a nilpotent endomorphism.

Still assuming that the eigenvalues of $u$ belong to $K$, we now apply Prop. 6 of VII, p. 34 to $u$. The polynomials $p_k$ are nothing other than the $X - a$ (as $a$ runs through the set of eigenvalues of $u$), and we see that $E$ is the direct sum of subspaces $E_i$ closed under $u$, cyclic (with respect to $u$), and such that the minimal polynomial of the restriction of $u$ to $E_i$ has the form $(X - a)^m$. Let $E_i'$ be the $K[X]$-module associated to $E_i$; then $E_i'$ is isomorphic to one of the modules $K[X]/((X - a)^m)$. Now the residue classes mod $(X - a)^m$ of the elements $(X - \alpha)^k$ ($0 \leq k \leq m - 1$) form a $K$-basis of $K[X]/((X - a)^m)$ (IV, p. 11, Cor.), and

for $0 \leq k \leq m - 1$; it follows that $E_i$ has dimension $m$, and if $a$ is the unique eigenvalue of the restriction $u_i$ of $u$ to $E_i$, then there exists a basis of $E_i$ with respect to which the matrix of $u_i$ is the $m \times m$ matrix