Page 161 - Textos de Matemática Vol. 39

STABILITY OF INVARIANT SUBSPACES 151
(iv) We have already seen that T−1XP ∈ S(T−1AT). Now we claim that ∀ε > 0 ∃δ > 0 such that if ∥ T−1AT −B ∥< δ then there is a matrix X′ ∈S(B)with∥T−1XP−X′ ∥<ε. ε
Letε>0andchoosearealnumberε1 suchthat0<ε1< ∥T−1∥∥P∥. Since X is A-stable, ∃δ1 > 0 such that if ∥ A−A′ ∥< δ1 then there is
amatrixX′ ∈S(A′)with∥X−X′ ∥<ε1.
Let 0 < δ < δ1 and assume that ∥ T−1AT −B ∥< δ.
Then
∥T−1 ∥∥T ∥
∥ A−TBT−1 ∥ = ∥ TT−1ATT−1 −TBT−1 ∥ ≤ ∥T∥∥T−1∥∥T−1AT−B∥
< ∥T∥∥T−1∥δ<δ1.
Thus there exists X′ ∈ S(TBT−1) such that ∥ X − X′ ∥< ε1. By item
(iii), T −1X′P ∈ S(B) and
∥T−1XP−T−1X′P∥ ≤ ∥T−1∥∥P∥∥X−X′∥
< ∥T−1∥∥P∥ε1<ε.
This proves that if X is A-stable then T−1XP is T−1AT-stable. The converse is an immediate consequence of the fact that A = T(T−1AT)T−1.
Following [2, Definition 1.2, p. 221] we give the following definition.
Definition 2.3. Let A ∈ Cn×n. If A1 ∈ Cp×p and A2 ∈ Cq×q are matrices
such that Λ(A1)∩Λ(A2) = ∅ and A is similar to Diag(A1,A2), then A1 ⊕A2 is
ofAandX∈S(A),anymatrixoftheform X1 0 ,withX1 ∈S(A1)and 0 X2
said to be a simple decomposition of A. If A ⊕ A is a simple decomposition 12
X2 ∈ S(A2), will be said to be a reduced form of X with respect to A1 ⊕ A2.
Notice that by item (iii) of Proposition 2.2 any X ∈ S(A) admits a reduced form provided that A1 ⊕ A2 is a simple decomposition of A.
Our next result is a slight modification of [2, Theorem 2.8, p. 238]. Symbol sep(A1,A2) appears in the statement. According to [2, p. 231] it is defined as follows: For A1 ∈ Cp×p and A2 ∈ Cq×q, XA1 − A2X = 0 is a Sylvester equation with a trivial solution if and only if Λ(A1) ∩ Λ(A2) = ∅. Thus, under this condition, XA1 −A2X ̸= 0 if and only if X ̸= 0. Using the operator norm
sep(A1,A2):= min ∥XA1−A2X∥2. ∥X∥2=1
Thus sep(A1, A2) > 0 if and only if Λ(A1) ∩ Λ(A2) = ∅.