My intuition of a nilpotent map is that it is a map for which some Vectors are cycled through all linearly independant vectors, until it hits one which is in the kernel, and then it collapses. The ones that can chain the longest without hitting a kernel vector are the ones which determine the $n \in \mathbb{N}_{0}$ such that $T^{n}(v) = 0 \quad \forall \; v \in V$. That way of seeing it comes from the following theorem :
- Let $n$ be as before and $u$ such that : $T^{k}(u) = 0 \; \; \land \; \; T^{k-1}(u) \neq 0$. The Set: $ \{T^{l}(u) \;| \; 0 \leq l < k\} $ is linearly independant.
Now by using this theorem for $ k = n $, you get the «longest chain» i was talking about. Now here is my question:
- Is that way of seing it a correct one ?
In some way it is seems quite similar to the way generalized eigenvectors work.