ConductorOfAVector 2003-12-02 11:57:27 2007-10-03 17:54:23 Definition % 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 vector space, $T:V\to V$ a linear transformation, and $W$ a $T$-invariant subspace of $V$. Let $x \in V$. The \emph{$T$-conductor} of $x$ \emph{in} $W$ is the set $S_T(x, W)$ containing all polynomials $g \in k[X]$ such that $g(T)x \in W$. It happens to be that this set is an ideal of the polynomial ring. We also use the term $T$-conductor of $x$ in $W$ to refer to the generator of such ideal. In the special case $W=\{0\}$, the $T$-conductor is called \emph{$T$-annihilator} of $x$. Another way to define the $T$-conductor of $x$ in $W$ is by saying that it is a monic polynomial $p$ of lowest degree such that $p(T)x \in W$. Of course this polynomial happens to be unique. So the $T$-annihilator of $x$ is the monic polynomial $m_x$ of lowest degree such that $m_x(T)x = 0$.