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nilpotent transformation

rmilson

2013-03-21 13:29:15

A linear transformationN :UUis called nilpotent if there exists ak Nsuch that

Nk = 0.

A nilpotent transformation naturally determines a flag of subspaces

{0} ker N1 ker N2 . . . ker Nk1 ker Nk =U

and a signature

0 =n0 < n1 < n2 < . . . < nk1 < nk = dim U, ni = dim ker Ni.

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

nj+1 nj nj nj1

for all j = 1, . . . , k 1. Equivalently, there exists a basis of U such that thematrix ofN relative to this basis is block diagonal

N1 0 0 . . . 0

0 N2 0 . . . 0

0 0 N3 . . . 0...

......

. . . ...

0 0 0 . . . N k

,

NilpotentTransformation created: 2013-03-21 by: rmilson version: 31961 Pri-vacy setting: 1 Definition 15-00

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with each of the blocks having the form

Ni =

0 1 0 . . . 0 00 0 1 . . . 0 0...

......

. . . ...

0 0 0 . . . 1 0

0 0 0 . . . 0 1

0 0 0 . . . 0 0

Lettingdi denote the number of blocks of sizei, the signature ofN is given by

ni = ni1+ di+di+1+. . .+dk, i= 1, . . . , k

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