Jordan blocks, Jordan strings, and their endpoints

For $m\ge 1$ and $\lambda \in F$, the Jordan block $J_m(\lambda )$ is the $m\times m$ matrix with $\lambda$ on the diagonal, $1$ on the superdiagonal, and $0$ elsewhere.

Let $T:V\to V$ be an endomorphism. A Jordan string of length $m$ for $T$ at $\lambda$ is an ordered list $(v_1,\dots ,v_m)$ satisfying $(T-\lambda I)v_1=0,(T-\lambda I)v_j=v_{j-1}(2\le j\le m)\mathrm{.}$ Its initial vector is $v_1$ and its terminal vector is $v_m$. The initial vector is required to be nonzero; consequently every vector in the string is nonzero. In the ordered basis $(v_1,\dots ,v_m)$, the restriction of $T$ to the string's span has matrix $J_m(\lambda )$ under the coordinate-column convention (Coordinate columns $[v{]}_{\mathcal{B}}$ and matrices $[T{]}_{\mathcal{B}}^{\mathcal{C}}$ of linear maps relative to ordered bases).

A Jordan string for a nilpotent endomorphism means a string at $\lambda =0$.