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 ${p}_{k1}$ 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)
Title  cyclic decomposition theorem 

Canonical name  CyclicDecompositionTheorem 
Date of creation  20130322 14:05:10 
Last modified on  20130322 14:05:10 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  16 
Author  CWoo (3771) 
Entry type  Theorem 
Classification  msc 15A04 
Synonym  Tadmissible 
Synonym  $T$admissible 
Related topic  CyclicSubspace 
Defines  admissible subspace 