16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 7.1.8

A cyclic vector $v$ means the orbit $\{v, Tv, T^2v, \ldots\}$ spans all of $V$—so $T$ is completely determined by how it acts on this single vector. The key insight is that any operator $U$ commuting with $T$ must respect this cyclic structure: if $U$ and $T$ commute, then $U$ preserves the span of powers of $T$ applied to $v$. This forces $U$ to act on the basis $\{v, Tv, T^2v, \ldots\}$ in a way that's entirely determined by where $U$ sends $v$, which must itself be a polynomial combination of these basis vectors.

A cyclic vector $v$ means the orbit $\{v, Tv, T^2v, \ldots\}$ spans all of $V$—so $T$ is completely determined by how it acts on this single vector. The key insight is that any operator $U$ commuting with $T$ must respect this cyclic structure: if $U$ and $T$ commute, then $U$ preserves the span of powers of $T$ applied to $v$. This forces $U$ to act on the basis $\{v, Tv, T^2v, \ldots\}$ in a way that's entirely determined by where $U$ sends $v$, which must itself be a polynomial combination of these basis vectors.