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[parent] cyclic decomposition theorem (Theorem)

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$ $ 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

  1. $ V=W_0\oplus Z(x_1,T)\oplus \cdots \oplus Z(x_r,T)$ (See the cyclic subspace definition)
  2. $ p_k$ divides $ p_{k-1}$ for every $ k=2,...,r$
Moreover, the integer $ r$ and the minimal polynomials $ 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)



"cyclic decomposition theorem" is owned by CWoo. [ full author list (2) | owner history (1) ]
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See Also: cyclic subspace

Other names:  T-admissible, $T$-admissible
Also defines:  admissible subspace

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Cross-references: integer, divides, cyclic subspace, annihilator polynomials, zero vectors, polynomial, subspace, linear operator, vector space, finite dimensional, field
There are 3 references to this entry.

This is version 13 of cyclic decomposition theorem, born on 2003-12-02, modified 2007-11-05.
Object id is 5449, canonical name is CyclicDecompositionTheorem.
Accessed 4103 times total.

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