Page 165 - Textos de Matemática Vol. 39

STABILITY OF INVARIANT SUBSPACES 155
A consequence of this theorem is that the set of matrices A ∈ Cn×n having a simple decomposition A1 ⊕ A2 with A1 ∈ Cp×p and A2 ∈ Cq×q is open. This set is also dense because it contains the set of diagonalizable matrices as a subset. Thus, the property of “having a simple decomposition” is a generic one. In addition, the simple decompositions of the matrices which are close enough to A are as close to the simple decompositions of A as one may desire.
We can give now the main result of this section. Theorem2.7.LetA∈Cn×n andX∈S(A).LetA ⊕A beasimplede-
X10 12
composition of A and X = 0 X2 a reduced form of X with respect to this
decomposition. Then X is A-stable if and only if Xi is Ai -stable, i = 1, 2. Proof. By Proposition 2.2 we can assume that A = A ⊕ A and
X1 0 1 2
X= 0 X2 .AlsobyProposition2.2,X∈S(A)ifandonlyifXi ∈S(Ai).
Assume first that Xi is Ai-stable for i = 1,2 and let ε > 0 be a given positive real number. This means that ∃ δi > 0 such that if ∥ Ai − A′i ∥< δi there is a matrix Xi′ ∈ S(A′i) satisfying ∥ Xi − Xi′ ∥< ε/2.
By Theorem 2.6, for δ3 = min{δ1 , δ2 } there exists δ > 0 such that if ∥ A−A′ ∥< δ then there is an invertible matrix T ∈ Cn×n such that T−1A′T = A′1⊕A′2 andmax{∥A1−A′1 ∥,∥A2−A′2 ∥}<δ3.Thatistosay,∥Ai−A′i ∥<δi,
i=1,2. Hence, for each i = 1,2 there exists X′ ∈ S(A′) with ∥ Xi −X′ ∥< ε/2. X′0ii i
We conclude that X′ = 1
0 X2′
is A′-invariant and ∥ X − X′ ∥< ε.
The converse is more involved. Assume that X is A-stable and let ε > 0
be a given positive real number.
(a) Forthisεthereisδ1 >0suchthatif∥X−X′ ∥<δ1 and∥Id−P ∥<δ1 then ∥ X − X′P ∥< ε.
(b) Let L ∈ Cd×d be the restriction of A to < X >. By Proposition 2.2 we can assume that L = L1 ⊕L2 and AiXi = XiLi, i = 1,2. For the real numberδ1 >0ofitem(a)andforρ>0suitableforA(andsoforL) there is δ2 > 0 such that if ∥ A−A′ ∥< δ2 then Λ(A′) ⊂ Vρ(A). And by Theorem 2.6, there is δ3 > 0 such that if ∥ L−L′ ∥< δ3 then an invertible matrix P ∈ Cd×d can be found so that P −1L′P = L′1 ⊕ L′2, max{∥ Id −P ∥,∥ L1 −L′1 ∥,∥ L2 −L′2 ∥} < δ1 and Λ(L′1) ⊂ Vρ(L1), Λ(L′2) ⊂ Vρ(L2).
(c) Since rankX = d, system XL = AX has a unique solution L = (X∗X)−1XAX. For the real number δ3 > 0 of item (b) there is δ4 >0suchthatif∥A−A′ ∥<δ4 and∥X−X′ ∥<δ4 thenrankX′ =d and the solution of system A′X′ = X′L′, L′ = (X′∗X′)−1X′A′X′, satisfies ∥ L − L′ ∥< δ3.