Page 170 - Textos de Matemática Vol. 39

160 ION ZABALLA
This implies that p(R) = 0. That is to say Rn−a1Rn−1−...−an=0 ⇒ zn(Rn−a1Rn−1−...−anId)=0
⇔ anzn=zn(Rn−1−a1Rn−2−...−an−1Id)=z1R. In conclusion, if L′ = R then A′X′ = X′L′ as desired.
We prove now that the nonderogatory matrices are the only ones with proper stable invariant subspaces. The proof is very similar to the corresponding one in [3].
Theorem 3.2. If A ∈ Cn×n is nilpotent and derogatory then the only stable invariant subspaces are {0} and Cn.
Proof. It is clear that both {0} and Cn are stable invariant subspaces of any matrix. Let 0 < d < n be a positive integer. We claim that there are not stable matrices in S(A).
As usual we can assume that A is in Jordan canonical form: ⎡⎢0 1 0 ··· 0⎤⎥
⎢0 0 1 ··· 0⎥
A = Diag(J1,...,Jr),Ji = ⎢. . . ... .⎥ ∈ Cni×ni
⎢⎣0 0 0 ··· 1⎥⎦ 000···0
withn1 ≥n2 ≥···≥nr >0.
IfX∈S(A),thereisauniqueL∈Cd×d suchthatAX=XL.This
equation allows us to parameterize the matrices X ∈ S(A). Put ⎡⎢ X 1 ⎤⎥ ⎡⎢ x i 1 ⎤⎥
(3.4) X=⎢X2⎥, X =⎢xi2 ⎥∈Cni×d ⎣.⎦ i ⎣.⎦
Xr xi,ni
where xij is the j-th row of Xi. Thus ⎡⎢ xi2 ⎤⎥ ⎡⎢ xi1L ⎤⎥
AX=XL⇔JiXi =XiL,i=1,...,r⇔⎢ . ⎥=⎢ .
0 xi,ni L
That is to say, xij = xi,j−1L = xi1Lj−1 with xi1Lni = 0. Therefore ⎡⎢ xi1 ⎤⎥
⎢ xi1L ⎥
( 3 . 5 ) X i = ⎢⎣ . . . ⎥⎦ , x i 1 L n i = 0 .
xi1 Lni −1
⎢xi3⎥ ⎢xi2L⎥
⎥. ⎢⎣xi,ni ⎥⎦ ⎢⎣xi,ni −1 L⎥⎦