# 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

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

and a signature

 $0=n_{0}

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

###### Proposition 1

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

 $n_{i}=n_{i-1}+d_{i}+d_{i+1}+\ldots+d_{k},\quad i=1,\ldots,k$
Title nilpotent transformation NilpotentTransformation 2013-03-22 12:19:52 2013-03-22 12:19:52 rmilson (146) rmilson (146) 7 rmilson (146) Definition msc 15-00 nilpotent LinearTransformation