nilpotent transformation

A linear transformation N:UU is called nilpotent if there exists a k such that


A nilpotent transformation naturally determines a flag of subspacesPlanetmathPlanetmathPlanetmath


and a signaturePlanetmathPlanetmathPlanetmathPlanetmath


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

Proposition 1

A sequencePlanetmathPlanetmath of increasing natural numbersMathworldPlanetmath is the signature of a nil-potent transformationMathworldPlanetmathPlanetmath if and only if


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


with each of the blocks having the form


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

Title nilpotent transformation
Canonical name NilpotentTransformation
Date of creation 2013-03-22 12:19:52
Last modified on 2013-03-22 12:19:52
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 7
Author rmilson (146)
Entry type Definition
Classification msc 15-00
Synonym nilpotent
Related topic LinearTransformation