Page 162 - Textos de Matemática Vol. 39

152 ION ZABALLA
Theorem 2.4. ([2, Theorem 2.8, p. 238]) Let A = A1 ⊕ A2 ∈ Cn×n be a
simple decomposition of A (i.e. A = Diag(A1,A2), A1 ∈ Cp×p, A2 ∈ Cq×q). Let E = E11 E12 ∈ Cn×n be partitioned according to the simple decomposition
E21 E22
of A; i.e., E11 ∈ Cp×p and E22 ∈ Cq×q. Let
γ=∥E21 ∥, η=∥E12 ∥, δ=sep(A1,A2)−∥E11 ∥−∥E22 ∥.
If δ > 0 and γη < 1 then there exists a matrix P ∈ Cq×p such that δ2 4
E21 + (A2 + E22)P − P(A1 + E11) − PE12P = 0. Moreover, for this matrix P the following properties hold:
2γ ≤2γ; δ + δ2 − 4γη δ
(i) ∥P ∥≤
( i i ) X˜ 1 = I p ∈ S ( A + E ) a n d Y˜ 2 = − P ∗ ∈ S ( A ∗ + E ∗ ) ;
P Iq
(iii) The restriction matrix of A + E to < X˜1 > is
A˜ 1 = A 1 + E 1 1 + E 1 2 P .
And the restriction matrix of A∗ + E∗ to < Y˜2 > is
A˜ ∗2 = A ∗2 + E 2∗ 2 − E 1∗ 2 P ∗ .
Remark 2.5. In [2, Theorem 2.11, p. 242] a procedure is given for constructing a uniquely determined matrix P satisfying the conditions of Theorem 2.4. The procedure is as follows: For matrix A + E it is proved that ([2, p. 234])
(2.1) sep(A1 +E11,A2 +E22)≥sep(A1,A2)−∥E11 ∥−∥E22 ∥.
Then a linear map is defined:
T : Cq×p → Cq×p
P P(A1 +E11)−(A2 +E22)P
According to (2.1) and taking into account the definition of sep, condition δ > 0 in Theorem 2.4 implies
inf ∥T(P)∥2>δ>0, ∥P ∥2
and this, in turn, implies that Λ(A1 + E11) ∩ Λ(A2 + E22) = ∅. Thus T is invertible. In these conditions inf ∥ T (P ) ∥2−1 = sup ∥ T −1(P ) ∥2, and so
−1 −1 ∥P∥2 ∥P∥2
∥ T ∥2 ≥ δ > 0. Now we can apply [2, Theorem 2.11, p. 242] recalling that we want to construct a matrix P satisfying E21 + (A2 + E22)P − P (A1 + E11) − PE12P = 0; i.e., T(P) = E21 − PE12P. The following sequence is defined:
Pk+1 =T−1(E21 −PkE12Pk)