cyclic subspace

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

Since $x,T(x),\mathrm{\dots },T^n(x),\mathrm{\dots }\in Z(x,T)$, we have that

$$W:=\mathrm{span}\{x,T(x),\mathrm{\dots },T^n(x),\mathrm{\dots }\}\subseteq Z(x,T).$$

On the other hand, since $W$ is $T$-invariant, $Z(x,T)\subseteq W$. Hence $Z(x,T)$ is the subspace generated by $x,T(x),\mathrm{\dots },T^n(x),\mathrm{\dots }$ In other words, $Z(x,T)=\{p(T)(x)\mid p\in k[X]\}$.

Remark. If $Z(x,T)=V$ we say that $x$ is a cyclic vector of $T$.

Title	cyclic subspace
Canonical name	CyclicSubspace
Date of creation	2013-03-22 14:05:03
Last modified on	2013-03-22 14:05:03
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	12
Author	CWoo (3771)
Entry type	Definition
Classification	msc 15A04
Classification	msc 47A16
Synonym	cyclic vector subspace
Related topic	CyclicDecompositionTheorem
Related topic	CyclicVectorTheorem
Defines	cyclic vector