# 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 $x,T(x),\ldots,T^{n}(x),\ldots\in Z(x,T)$, we have that

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

On the other hand, since $W$ is $T$-invariant, $Z(x,T)\subseteq W$. Hence $Z(x,T)$ is the subspace generated by $x,T(x),\ldots,T^{n}(x),\ldots$ 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 CyclicSubspace 2013-03-22 14:05:03 2013-03-22 14:05:03 CWoo (3771) CWoo (3771) 12 CWoo (3771) Definition msc 15A04 msc 47A16 cyclic vector subspace CyclicDecompositionTheorem CyclicVectorTheorem cyclic vector