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nilpotent transformation
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(Definition)
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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 $$0 = n_0 < n_1 < n_2 < \ldots < n_{k-1} < n_k = \dim U,\qquad n_i = \dim \ker N^i.$$ The signature is governed by the following constraint, and
characterizes $N$ up to linear isomorphism.
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 $$n_i = n_{i-1} + d_i + d_{i+1} + \ldots + d_k,\quad i=1,\ldots,k$$
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"nilpotent transformation" is owned by rmilson.
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Cross-references: size, number, diagonal, block, matrix, basis, transformation, natural numbers, increasing, sequence, linear isomorphism, signature, subspaces, flag, linear transformation
There are 6 references to this entry.
This is version 4 of nilpotent transformation, born on 2002-02-15, modified 2004-05-04.
Object id is 1961, canonical name is NilpotentTransformation.
Accessed 10201 times total.
Classification:
| AMS MSC: | 15-00 (Linear and multilinear algebra; matrix theory :: General reference works ) |
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Pending Errata and Addenda
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