PlanetMath: nilpotent transformation

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nilpotent transformation

(Definition)

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

$\displaystyle N^k = 0.$

A nilpotent transformation naturally determines a flag of subspaces

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

and a signature

$\displaystyle 0 = n_0 < n_1 < n_2 < \ldots < n_{k-1} < n_k = \dim U,\qquad n_i = \dim \ker N^i.$

The signature is governed by the following constraint, and characterizes

up to linear isomorphism.

Proposition 1 A sequence of increasing natural numbers is the signature of a nil-potent transformation if and only if

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

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

$\displaystyle \begin{pmatrix} N_1 & 0 & 0 & \ldots & 0\ 0 & N_2 & 0 & \ldots ... ...& \vdots & \vdots & \ddots & \vdots \ 0 & 0 & 0 & \ldots & N_k \end{pmatrix},$

with each of the blocks having the form

$\displaystyle N_i = \begin{pmatrix} 0 & 1 & 0 & \ldots & 0 & 0 \ 0 & 0 & 1 & ... ...& 0 \ 0 & 0 & 0 & \ldots & 0 & 1 \ 0 & 0 & 0 & \ldots & 0 & 0 \end{pmatrix}$

Letting denote the number of blocks of size , the signature of is given by

$\displaystyle n_i = n_{i-1} + d_i + d_{i+1} + \ldots + d_k,\quad i=1,\ldots,k$

"nilpotent transformation" is owned by rmilson.

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