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conductor of a vector (Definition)

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$.



"conductor of a vector" is owned by CWoo. [ full author list (2) | owner history (1) ]
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Other names:  T-conductor, conductor, annihilator, annihilator polynomial, conductor polynomial
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Cross-references: degree, monic polynomial, generator, term, polynomial ring, ideal, polynomials, subspace, linear transformation, vector space, field
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This is version 2 of conductor of a vector, born on 2003-12-02, modified 2007-10-03.
Object id is 5455, canonical name is ConductorOfAVector.
Accessed 7750 times total.

Classification:
AMS MSC: 15A04 (Linear and multilinear algebra; matrix theory :: Linear transformations, semilinear transformations)
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