A nilpotent operator is one Jordan block exactly when its kernel grows by one dimension at each power.Suggest a slogan
Let \(k\) be a field, \(V\) a \(k\)-vector space of dimension \(n \coloneqq \dim_k V \geq 1\), and \(T\colon V\to V\) a nilpotent linear map. The following are equivalent:
There exists a basis \(e_1, \ldots, e_n\) of \(V\) such that \[ T(e_i) = \begin{cases} e_{i-1} & \text{if } i \geq 2, \\ 0 & \text{if } i=1. \end{cases} \]
There exists a cyclic vector \(v \in V\): the vectors \(T^k v\) for \(k = 0, 1, \ldots, n-1\) form a basis of \(V\).
\(T^{n-1} \neq 0\).
\(\dim_k \ker T^\ell = \ell\) for each \(1\leq \ell \leq n\).
\(\dim_k \ker T = 1\).