Page 23 - Textos de Matemática Vol. 36

2.1. Solutions of ARE and graph subspaces 13
Notice that (2.6) means that R := M11 + M12W is the matrix of the restriction of the map, defined by M, to G(W) with respect to the basis defined by the
In
columns of the matrix W ; consequently this map has the same eigenvalues
as R.
Proof. If G(W) is M-invariant, there exists a matrix R such that
In In M W = W R.
The first and second part of this identity yield R = M11 + M12W
and
M21 +M22W(=WR)=WM11 +WM12W.
Therefore W is a solution of ARE. On the other hand, if W fulfills ARE, it follows by performing the first steps of the proof in the opposite direction that (2.6) holds with R := M11 + M12 W . This proves the assertions of the theorem.

Since every M-invariant subspace of M is spanned by generalized eigen- vectors of M, it follows from Theorem 2.1.2:
Corollary 2.1.3. For every solution W ∈ LARE there exist n generalized eigenvec- z
tors vνj = νj of M such that with Z = (zν1,...,zνn) ∈ Cn×n, yνj
Y = (yν1,...,yνn) ∈ Cm×n, we have
det Z ̸= 0 (2.7)
and
W = Y Z−1. (2.8)
Remark 2.1.4. (i) The solution W defined in Corollary 2.1.3 is called the so- lution of ARE corresponding to vν1,...,vνn. If vνj , 1 ≤ j ≤ n, corresponds to the eigenvalue λj , we also say that W is one of the solutions (or, if this is unique, the solution) corresponding to λν1 , . . . , λνn .
. There-
(ii) If (2.7) holds, then
Im Y +Im I =C ,
Z ˙ 0n×m m+n m
Z 0n×m
i.e., the subspace Im Y is complementary to Im I
m
fore (2.7) is called the complementarity condition (or disconjugacy condi- tion).