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

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