# conductor of a vector

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 *$T$-conductor* of $x$ *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 *$T$-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 ${m}_{x}(T)x=0$.

Title | conductor of a vector |
---|---|

Canonical name | ConductorOfAVector |

Date of creation | 2013-03-22 14:05:19 |

Last modified on | 2013-03-22 14:05:19 |

Owner | CWoo (3771) |

Last modified by | CWoo (3771) |

Numerical id | 5 |

Author | CWoo (3771) |

Entry type | Definition |

Classification | msc 15A04 |

Synonym | T-conductor |

Synonym | conductor |

Synonym | annihilator |

Synonym | annihilator polynomial |

Synonym | conductor polynomial |