Cyclic subspace

In mathematics, in linear algebra, a cyclic subspace is a certain special subspace of a finite-dimensional vector space associated with a vector in the vector space and a linear transformation of the vector space. The cyclic subspace associated with a vector v in a vector space V and a linear transformation T of V is called the T-cyclic subspace generated by v. The concept of a cyclic subspace is a basic component in the formulation of the cyclic decomposition theorem in linear algebra.

Let ${\displaystyle T:V\rightarrow V}$ be a linear transformation of a vector space ${\displaystyle V}$ and let ${\displaystyle v}$ be a vector in ${\displaystyle V}$. The ${\displaystyle T}$-cyclic subspace of ${\displaystyle V}$ generated by ${\displaystyle v}$ is the subspace ${\displaystyle W}$ of ${\displaystyle V}$ generated by the set of vectors ${\displaystyle \{v,T(v),T^{2}(v),\ldots ,T^{r}(v),\ldots \}}$. This subspace is denoted by ${\displaystyle Z(v;T)}$. If ${\displaystyle V=Z(v;T)}$, then ${\displaystyle v}$ is called a cyclic vector for ${\displaystyle T}$.

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