Page 167 - Textos de Matemática Vol. 39

STABILITY OF INVARIANT SUBSPACES 157
to characterize only the stable subspaces of nilpotent endomorphisms. This will be accomplished in the following section.
3. The nilpotent case
We start by characterizing the subspaces of nilpotent nonderogatory linear operators. Recall that nonderogatory means that there is only one Jordan block associated to each eigenvalue. Thus, nilpotent and nonderogatory is equivalent to being similar to a matrix with the following form
⎡⎢0 ··· 0 0⎤⎥ ⎢1 ··· 0 0⎥
⎢⎣ . . . . . . . . . . . . ⎥⎦ 0···10
An important property of these matrices is that there is a vector b ∈ Cn such that
Cn =< b,Ab,...,An−1b > . These vectors are sometimes called cyclic vectors (see [1])
Theorem 3.1. If A∈Cn×n is nilpotent and nonderogatory then all A-invariant subspaces are stable.
Proof. The goal is to prove that if X ∈ S(A) then X is stable. By Theorem 1.3 this will prove that any A-invariant subspace is stable as claimed.
LetL∈Cd×d betherestrictionofAto<X>.AsAisnilpotentand nonderogatory so is L. By Proposition 2.2 we can assume that A and L are in Jordan canonical form
⎡⎢0 ··· 0 0⎤⎥
⎢1 ··· 0 0⎥ n×n A = ⎢⎣ . . . . . . . . . . . . ⎥⎦ ∈ C .
0···10 and L like A but of size d×d. Thus
⎡⎢x1L⎤⎥ ⎣ . ⎦ ⎣ . ⎦
⎡⎢ 0 ⎤⎥
AX=⎢x1 ⎥ and XL=⎢x2L⎥.
xn−1 xn L where xi is the i-th row of X.