PlanetMath: cyclic subspace

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cyclic subspace

(Definition)

Let be a vector space over a field , and $x \in V$ . Let $T:V\to V$ be a linear transformation. The -cyclic subspace generated by is the smallest -invariant subspace which contains , and is denoted by .

Since $x,T(x),\ldots, T^n(x),\ldots \in Z(x,T)$ , we have that

$\displaystyle W:=\operatorname{span}\lbrace x,T(x),\ldots,T^n(x),\ldots\rbrace \subseteq Z(x,T).$

On the other hand, since

-invariant, $Z(x,T)\subseteq W$ . Hence

is the subspace generated by $x,T(x),\ldots, T^n(x),\ldots$ In other words, $Z(x,T)=\{p(T)(x) \mid p \in k[X]\}$ .

Remark. If we say that is a cyclic vector of .

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

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Other names:

cyclic vector subspace

Also defines:

cyclic vector

This object's parent.

Attachments:: cyclic decomposition theorem (Theorem) by CWoo
cyclic vector theorem (Theorem) by gumau
theorem about cyclic subspaces (Theorem) by Mathprof

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Cross-references: contains, generated by, subspace, linear transformation, field, vector space
There are 5 references to this entry.

This is version 9 of cyclic subspace, born on 2003-12-02, modified 2007-10-03.
Object id is 5447, canonical name is CyclicSubspace.
Accessed 5330 times total.

Classification:

AMS MSC:	15A04 (Linear and multilinear algebra; matrix theory :: Linear transformations, semilinear transformations)
	47A16 (Operator theory :: General theory of linear operators :: Cyclic and hypercyclic vectors)

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