Page 160 - Textos de Matemática Vol. 39

150 ION ZABALLA
Assume that X is A-stable and let ε > 0. There exists δ > 0 such thatif∥A−A′ ∥<δthenthereisX′ ∈S(A′)suchthat∥X−X′ ∥<ε. We aim to see that if ∥ λ0In −A−B′ ∥< δ then there exists X′ ∈ S(B′) such that ∥ X − X′ ∥< ε. This will prove that X is (λ0In − A)-stable. But ∥ λ0In −A−B′ ∥=∥ (λ0In −B′)−A ∥ and since X is A-stable, there exists X′ ∈ S(λ0In − B′) such that ∥ X − X′ ∥< ε. Our claim follows from the fact that S(λ0In − B′) = S(B′).
In the sequel Λ(A) will denote the spectrum of the square complex matrix A; i.e. the set of eigenvalues of A. An immediate consequence of the above Proposition is that the stability of the A-invariant subspaces with Λ(A) = {λ0} can be reduced to the stability of the A-invariant subspaces with Λ(A) = {0}; i.e. A nilpotent. The remainder of this section is devoted to prove that we can assume without loss of generality that A only has one eigenvalue. We start with the following basic result:
Proposition 2.2. Let A ∈ Cn×nand Λ(A) = Λ1 ∪ Λ2 with Λ1 ∩ Λ2 = ∅. Let X∈S(A)andletL∈Cd×d betherestrictionmatrixofAto<X>1.Then there are invertible matrices T ∈ Cn×n and P ∈ Cd×d for which the following properties hold: A 0
(i) T−1AT = (ii) P−1LP =
1 , Λ(A1) = Λ1 and Λ(A2) = Λ2; 0 A2
L1 0 , Λ(L1) ⊆ Λ(A1) and Λ(L2) ⊆ Λ(A2); 0 L2 X 0
(iii) T−1XP ∈ S(T−1AT) and T−1XP = 1 with A1X1 = X1L1
0 X2
(iv) X is A-stable if and only if T−1XP is (T−1AT)-stable.
Proof. (i) This is a very well-known property.
(ii) The proof is like in (i) bearing in mind that since L is the restriction
matrix of A to < X >, Λ(L) ⊆ Λ(A) = Λ1 ∪ Λ2.
(iii) FirstT−1XP ∈S(T−1AT)becauseT−1ATT−1XP =T−1XPP−1LP.
Writing this identity explicitly,
A1 0 X1 Y1 = X1 Y1 L1 0 ,
0 A2 Y2 X2 Y2 X2 0 L2
we have that A1X1 = X1L1, A2X2 = X2L2, A1Y1 = Y1L2 and A2Y2 = Y2L1. As Λ(A1)∩Λ(L2) = Λ(A2)∩Λ(L1) = ∅, we conclude that Y1 = 0 and Y2 = 0 (the sizes of these two zero matrices may be different from each other).
1It must be understood that L is the unique solution of the equation AX = XL
and A2X2 = X2L2;