|
|
|
|
![[parent]](http://images.planetmath.org:8080/images/uparrow.png)
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
- $V=W_0\oplus Z(x_1,T)\oplus \cdots \oplus Z(x_r,T)$ (See the cyclic subspace definition)
- $p_k$ 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) ]
|
|
(view preamble | get metadata)
See Also: cyclic subspace
| Other names: |
T-admissible,  -admissible |
| Also defines: |
admissible subspace |
This object's parent.
|
|
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 5631 times total.
Classification:
| AMS MSC: | 15A04 (Linear and multilinear algebra; matrix theory :: Linear transformations, semilinear transformations) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|
