cyclic decomposition theorem

Let $k$ be a field, $V$ a finite dimensional vector space over $k$ and $T$ a linear operator over $V$. Call a subspace $W\subseteq V$ $T$-admissible if $W$ is $T$-invariant and for any polynomial $f(X)\in k[X]$ with $f(T)(v)\in W$ for $v\in V$, there is a $w\in W$ such that $f(T)(v)=f(T)(w)$.

Let $W_0$ be a proper $T$-admissible subspace of $V$. There are non zero vectors $x_1,\mathrm{\dots },x_r$ in $V$ with respective annihilator polynomials $p_1,\mathrm{\dots },p_r$ such that

1. $V=W_0\oplus Z(x_1,T)\oplus \mathrm{\cdots }\oplus Z(x_r,T)$ (See the cyclic subspace definition)

2. $p_k$ divides $p_{k-1}$ for every $k=2,\mathrm{\dots },r$

Moreover, the integer $r$ and the minimal polynomials (http://planetmath.org/MinimalPolynomialEndomorphism) $p_1,\mathrm{\dots },p_r$ are uniquely determined by (1),(2) and the fact that none of $x_k$ is zero.

This is “one of the deepest results in linear algebra” (Hoffman & Kunze)