conductor of a vector

Let k be a field, V a vector spaceMathworldPlanetmath, T:VV a linear transformation, and W a T-invariant subspace of V. Let xV. The T-conductor of x in W is the set ST(x,W) containing all polynomialsMathworldPlanetmathPlanetmath gk[X] such that g(T)xW. 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 generatorPlanetmathPlanetmathPlanetmath of such ideal.

In the special case W={0}, the T-conductor is called T-annihilatorPlanetmathPlanetmath 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)xW. Of course this polynomial happens to be unique. So the T-annihilator of x is the monic polynomial mx of lowest degree such that mx(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