nilpotent transformation

A linear transformation $N:U\rightarrow U$ is called nilpotent if there exists a $k\in\mathbb{N}$ such that

N^{k}=0.

A nilpotent transformation naturally determines a flag of subspaces

\{0\}\subset\ker N^{1}\subset\ker N^{2}\subset\ldots\subset\ker N^{k-1}\subset% \ker N^{k}=U

0=n_{0}<n_{1}<n_{2}<\ldots<n_{k-1}<n_{k}=\dim U,\qquad n_{i}=\dim\ker N^{i}.

The signature is governed by the following constraint, and characterizes $N$ up to linear isomorphism.

A sequence of increasing natural numbers is the signature of a nil-potent transformation if and only if

n_{j+1}-n_{j}\leq n_{j}-n_{j-1}

for all $j=1,\ldots,k-1$ . Equivalently, there exists a basis of $U$ such that the matrix of $N$ relative to this basis is block diagonal

\begin{pmatrix}N_{1}&0&0&\ldots&0\\ 0&N_{2}&0&\ldots&0\\ 0&0&N_{3}&\ldots&0\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 0&0&0&\ldots&N_{k}\end{pmatrix},

with each of the blocks having the form

N_{i}=\begin{pmatrix}0&1&0&\ldots&0&0\\ 0&0&1&\ldots&0&0\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 0&0&0&\ldots&1&0\\ 0&0&0&\ldots&0&1\\ 0&0&0&\ldots&0&0\end{pmatrix}

Letting $d_{i}$ denote the number of blocks of size $i$ , the signature of $N$ is given by

n_{i}=n_{i-1}+d_{i}+d_{i+1}+\ldots+d_{k},\quad i=1,\ldots,k