# 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 $x_{1},...,x_{r}$ in $V$ with respective annihilator polynomials $p_{1},...,p_{r}$ such that

1. 1.

$V=W_{0}\oplus Z(x_{1},T)\oplus\cdots\oplus Z(x_{r},T)$ (See the cyclic subspace definition)

2. 2.

$p_{k}$ divides $p_{k-1}$ for every $k=2,...,r$

Moreover, the integer $r$ and the minimal polynomials (http://planetmath.org/MinimalPolynomialEndomorphism) $p_{1},...,p_{r}$ 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)

Title cyclic decomposition theorem CyclicDecompositionTheorem 2013-03-22 14:05:10 2013-03-22 14:05:10 CWoo (3771) CWoo (3771) 16 CWoo (3771) Theorem msc 15A04 T-admissible $T$-admissible CyclicSubspace admissible subspace