nilpotent transformation
A linear transformation N:U→U is called nilpotent if there exists a k∈ℕ such that
Nk=0. |
A nilpotent transformation naturally determines a flag of subspaces
{0}⊂kerN1⊂kerN2⊂…⊂kerNk-1⊂kerNk=U |
and a signature
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
for all . Equivalently, there exists a basis of
such that the matrix of relative to this basis is block diagonal
with each of the blocks having the form
Letting denote the number of blocks of size , the signature of 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 |