Page 47 - Textos de Matemática Vol. 36

2.4. Flows on Graßmann manifolds 37
Together with Section 2.1 we have now three different possibilities (see also Remark 2.3.2 (iii) to determine the set Γ of all (complex) solutions of equation ARE:
−1 Q (m+n)×n Q
(i) Γ={PQ | P ∈C ,detQ̸=0 and Im P isM-invariant}
(see Theorem 2.1.2).
(ii) Γ = Cm×n ∩ {W (eiφ∞; WC )|C ∈ C(m+n)×n, det Q0(C) ̸= 0 and
φ ∈/ {φ1,...,φ2h}}.
This is an immediate consequence of Lemma 2.4.1 and the representation formula (2.30). If the limit W(eiφ∞;WC) exists, then from a general re- sult on differential equations (see [KnKa74]) it follows that it must be a stationary point of the differential equation i.e., a solution of the ARE. On the other hand if W0 ∈ Γ is any solution of ARE, then we may choose
0
. Then (2.30) represents that constant solution W0 with
(iii) Γ = { all constant Cm×n -valued functions W (·; WC ) of the form (2.30)}. Hence, Γ denotes the set of all solutions of RDE such that by an ap- propriate choice of the parameter matrix C in (2.30) all matrix elements wlα ∈C, 1≤l≤m, 1≤α≤n,becomeconstant.
Using (i) – (iii) and the fundamental representation formula (2.30) we obtain a useful parameterization of Γ.
Therefore we introduce some abbreviations. Let Vn(C(m+n)×n) be the set of all full rank (m + n) × n matrices with complex entries. For a matrix C ∈ Vn(C(m+n)×n) and 1 ≤ k ≤ N we define the set of indices
J0(C):={j ∈N|1≤j ≤N and D(aj,C)̸=0} and the set of parameter matrices
Gζ(ak) :=
{C ∈ Vn(C(m+n)×n)|D(aν,C) = 0, for ζ(aν) ̸= ζ(ak) and D(ak,C) ̸= 0}.
Notice that for all C ∈ Gζ(ak) the exponential terms in (2.30) are identical to etζ(ak). Hence, for those C ∈ Vn(C(m+n)×n), (2.30) becomes
vlα(aν)D(aν,C)
ν∈J0(C)
wlα(t, WC ) = v˜(aν )D(aν , C) , (2.33)
(m+n)×n In suchthatin(2.12)att=0wehaveVC= W ,i.e.,
C∈C
C = V W
−1 In 0
the same limit.

ν∈J0(C)