Page 24 - Textos de Matemática Vol. 36

14 (iii)
Chapter 2. Global aspects
Theorem 2.1.2 and Corollary 2.1.3 mean that each W ∈ LARE corresponds to a unique n-dimensional, M-invariant subspace satisfying the comple- mentarity condition. This fact can be used to determine LARE and its structure.
(iv)
Example 2.1.5. Consider the HARE
with
i.e.,
ThereexistalsoinfinitedimensionalversionsofTheorem2.1.2(see[LaTr91], [CDW95], [KuZw96], [GaHo99]).
−A∗X − XA − Q + XSX = 0 00 10 11
A=00,Q=00,S=11,
 0 0 −1 −1  H=0 0−1−1.
 − 1 0 0 0  0000
H has the simple eigenvalues 1 and −1 and the eigenvalue 0 of algebraic and geometric multiplicity 2.
1 1 0 0 v−1=1, v1= 1 , v01=1 and v02= 0 
 1   − 1   0   1  0 0 0 −1
define a corresponding Jordan basis of eigenvectors. Consequently there exist two 1-parametric families of 2-dimensional, H-invariant graph subspaces:
10 10 S1(α)=Im 1 1  and S2(α)=Im 1 1 , α∈C.
 1 α   − 1 α  0 −α 0 −α
Therefore, by Corollary 2.1.3, there are two families of solutions of HARE; these have the form
1−α α −1−α α
X1(α) = α −α and X2(α) = α −α , α ∈ C
and are Hermitian if and only if α ∈ R. This shows that LHARE and HHARE are both uncountable and unbounded.