cyclic decomposition theorem CyclicDecompositionTheorem 2003-12-02 05:06:39 2007-11-05 01:12:43 Theorem cyclic subspace admissible subspace % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here Let $k$ be a field, $V$ a finite dimensional vector space over $k$ and $T$ a linear operator over $V$. Call a subspace $W\subseteq V$ \emph{$T$-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 \begin{enumerate} \item $V=W_0\oplus Z(x_1,T)\oplus \cdots \oplus Z(x_r,T)$ (See the cyclic subspace definition) \item $p_k$ divides $p_{k-1}$ for every $k=2,...,r$ \end{enumerate} Moreover, the integer $r$ and the \PMlinkname{minimal polynomials}{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)