Cyclic vector ($V=\mathrm{span}\{v,Tv,T^{2}v,\dots \}$)

For a linear operator $T$, a cyclic vector is a vector whose iterates span the entire vector space. In finite dimension $d$, the vectors $v,Tv,\dots ,T^{d-1}v$ then form a basis: the first linear dependence makes the preceding span invariant under $T$, so later iterates cannot enlarge it. In topological settings one often requires the iterates to have dense span instead; that closure convention must be stated.

On a $d$-dimensional vector space, a cyclic vector $v$ gives a basis $v,Tv,\dots ,T^{d-1}v$. Write $T^{d}v=-{\sum }_{j<d}{c}_j{T}^{j}v$ and set $f(t)=t^d+{\sum }_{j<d}{c}_j{t}^j$. Then $f(T)v=0$. Since polynomials in $T$ commute with $T$, $f(T)$ kills every basis vector, so $f(T)=0$. No nonzero polynomial of degree less than $d$ can kill $v$, by linear independence. Hence the minimal polynomial is $f$. The matrix in this basis is $C(f)$, and the characteristic polynomial of a companion matrix is $f$ as well.

A cyclic subspace generated by $x$ under a linear operator $T$ is the span of its successive iterates $x,Tx,T^{2}x,\dots$. In finite dimension it is already generated by the first $n$ iterates. The first linear dependence expresses the next iterate in previous ones and proves invariance without using the Cayley-Hamilton theorem.