Page 163 - Textos de Matemática Vol. 39

then
T h u s , i f w e p u t A˜ = A + E , t h e n
STABILITY OF INVARIANT SUBSPACES 153
with P0 = 0. This sequence converges and its (unique) limit, P , satisfies con- dition (i) of Theorem 2.4. Conditions (ii) and (iii) are consequences of the equation E21 + (A2 + E22)P − P(A1 + E11) − PE12P = 0.
For the next theorem it is important to bear in mind the following conse- quences of Theorem 2.4. If properties (ii) and (iii) of Theorem 2.4 are satisfied
for some matrix P and
T = I p 0 = X˜ 1 X˜ 2 , P Iq
T − 1 = I p 0 = Y˜ 1 ∗ . −P Iq Y˜2∗
T − 1 A˜ T = Y˜ 1 ∗ A˜ X˜ 1 Y˜ 1 ∗ A˜ X˜ 2 . Y˜ 2 ∗ A˜ X˜ 1 Y˜ 2 ∗ A˜ X˜ 2
Now, according to condition (ii) of Theorem 2.4 < A˜X˜1 >⊂< X˜1 > and by the definition of Y˜2∗ and X˜1, Y˜2∗X˜1 = 0 . Therefore Y˜2∗A˜X˜1 = 0.
Furthermore, as X˜1 ∈ S(A˜) and A˜1 is the restriction of A˜ to < X˜1 >
Y˜ 1 ∗ A˜ X˜ 1 = Y˜ 1 ∗ X˜ 1 A˜ 1 = I p 0 I p A˜ 1 = A˜ 1 . P
Similarly
In conclusion
Y˜2∗A˜X˜2 = A˜2Y˜2∗X˜2 = A˜2. T − 1 A˜ T = A˜ 1 E 1 2 .
Given a real number ρ > 0, Bρ(λ) is the open ball with center at λ and radius ρ. For a given square matrix A, if Λ(A) = {λ1,...,λv} is its spectrum, the ρ-neighbourhood of the spectrum of A is defined as
v i=1
provided that the open balls are pairwise disjoint. A real number ρ, small enough as to satisfy this property, is said to be suitable for A.
In the sequel we will assume that the matrix norm that we use is the l1 norm as defined in the introduction. Actually, most properties will held true for any matrix norm, but in some steps working with the l1 norm makes the computations easier to follow.
Vρ(A) :=
Bρ(λi)
0 A˜2