Page 166 - Textos de Matemática Vol. 39

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ION ZABALLA
(d) Let δ5 = min{δ1,δ4}. For this δ5, and since X is A-stable, there is δ6 > 0 such that if ∥ A−A′ ∥< δ6 then there exits X′ ∈ S(A′) such that ∥X − X′∥ < δ5.
Now we put all this together. Let δ = 1 min{δ2,δ4,δ6} and assume that 2
∥Ai−A′i∥<δfori=1,2.LetA′ =A′1⊕A′2.Thus∥A−A′ ∥<2δ= min{δ2,δ4,δ6}. By item (d), there exists X′ ∈ S(A′) such that ∥ X−X′ ∥< δ5. As X′ ∈ S(A′), there exits a unique matrix L′ such that A′X′ = X′L′; and since∥A−A′ ∥<δ4 and∥X−X′ ∥<δ5 ≤δ4,byitem(c),∥L−L′ ∥<δ3.Now, by item (b) and for a ρ > 0 suitable for A (and for L) we have that: on the one hand,as∥A−A′ ∥<δ≤δ2,Λ(A′)⊂Vρ(A).ButA′ =A′1⊕A′2 andsoΛ(A′i)⊂ Vρ(Ai), i = 1,2. On the other hand, there is an invertible matrix P ∈ Cd×d s u c h t h a t P − 1 L ′ P = L ′1 ⊕ L ′2 , m a x { ∥ I d − P ∥ , ∥ L 1 − L ′1 ∥ , ∥ L 2 − L ′2 ∥ } < δ 1 and Λ(L′1) ⊂ Vρ(L1), Λ(L′2) ⊂ Vρ(L2). Finally, as ∥ X − X′ ∥< δ5 ≤ δ1 and ∥Id −P ∥<δ1,byitem(a),∥X−X′P ∥<ε.
0 A′2 Y2′ X2′ Y2′ X2′ 0 L′2
As X′ ∈ S(A′) we have that X′P ∈ S(A′) too. In addition A′X′ = X′L′ i m p l i e s t h a t A ′ X ′ P = X ′ P P − 1 L ′ P . T h a t i s t o s a y , i f X ′ P = X 1′ Y 1 ′ t h e n
Y2′ X2′ A ′1 0 X 1′ Y 1′ = X 1′ Y 1′ L ′1 0 .
Hence A′1X1′ = X1′ L′1, A′2X2′ = X2′ L′2, A′1Y1′ = Y1′L′2 and A′2Y2′ = Y2′L′1. But Λ(A′1) ⊂ Vρ(A1), Λ(A′2) ⊂ Vρ(A2), Λ(L′1) ⊂ Vρ(L1), Λ(L′2) ⊂ Vρ(L2). More- over Λ(L1) ⊂ Λ(A1), Λ(L2) ⊂ Λ(A2) and Λ(A1) ∩ Λ(A2) = ∅. In conclusion Λ(A′1) ∩ Λ(L′2) = Λ(A′2) ∩ Λ(L′1) = ∅. This implies that the only solutions of the equations A′1Y1′ = Y1′L′2 and A′2Y2′ = Y2′L′1 are the trivial ones: Y1′ = 0 and Y2′ =0.ThereforeXi′ ∈S(A′i),i=1,2andε>∥X−X′P∥=∥X1−X1′ ∥+ ∥X2−X2′ ∥.Thus∥Xi−Xi′∥<εfori=1,2andthetheoremfollows.
If A ∈ Cn×n and Λ(A) = {λ1,...,λv} is the spectrum of A then the subspace Ker(λiIn − A)n is called the root subspace of A associated to λi. It is well-known that if Ai is a restriction matrix of A to Ker(λiIn − A)n then A1 ⊕ A2 ⊕ ··· ⊕ Av is a simple decomposition of A and Λ(Ai) = {λi}. If X ∈ S(A) and Diag(X1,...,Xv) is a reduced form of X with respect to A1 ⊕ A2 ⊕ ··· ⊕ Av we have, by Theorem 2.7, that X is A-stable if and only if Xi is Ai-stable, i = 1,...,v. So we can reduce the study of the stability of invariant subspaces to matrices with only one eigenvalue. The root subspace of such a matrix is the whole space Cn. And bearing in mind Proposition 2.1 we conclude that we can reduce our analysis to nilpotent matrices.
Another consequence of Theorem 2.7 is that if A1 ⊕ A2 ⊕ · · · ⊕ Av is a simple decomposition corresponding to the root subspaces of A then the A-stable subspaces are direct sums of the Ai-stable subspaces. Thus, we have