nilpotent transformation

A linear transformation $N:U\to U$ 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

$$\{0\}\subset \ker N^1\subset \ker N^2\subset \mathrm{\dots }\subset \ker N^{k-1}\subset \ker N^k=U$$

and a signature

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

Proposition 1

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

$$n_{j+1}-n_j\le n_j-n_{j-1}$$

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

with each of the blocks having the form

Letting $d_i$ denote the number of blocks of size $i$, thesignature of $N$ is given by

$$n_i=n_{i-1}+d_i+d_{i+1}+\mathrm{\dots }+d_k,i=1,\mathrm{\dots },k$$