Page 53 - Textos de Matemática Vol. 36

2.4. Flows on Graßmann manifolds
43
Theorem 2.4.6. Let R(ak) ̸= R(aj) for j ̸= k and v˜(ak) ̸= 0. (i) If C ∈ Gζ(ak) then WC := Wk, with Wk = (wk )1≤l≤m
defined by
(2.37) for1≤l≤m, 1≤α≤n,isasolutionofequationARE.Furthermore
Wk is real if M is a real matrix. Obviously WC does not depend on C here. (ii) If
C1 ∈ {C ∈ Vn(C(m+n)×n)|D(ak,C) ̸= 0 and D(aj,C) = 0 if
R(aj) > R(ak) or if R(aj) = R(ak) with ζ(aj) ̸= ζ(ak)}
then there exists t0 ∈R, K >0 such that W(t;WC1)∈Cm×n for t>t0 and
|w (t,W 1)−wk | ≤ Ke(R(ak−1)−R(ak))t, 1 ≤ l ≤ m, 1 ≤ α ≤ n, (2.38) lα C lα
which means that the convergence of W(t,WC1) → Wk, as t → ∞, takes place at an exponential rate.
Remark 2.4.7. As in the preceding theorem let R(aj) ̸= R(ak),j ̸= k, but v˜(ak) = 0.
If
C2 ∈ {C ∈ Vn(C(m+n)×n)|D(ak,C) ̸= 0 and D(aj,C) = 0 if
R(aj)>R(ak) orif R(aj)=R(ak) with ζ(aj)̸=ζ(ak)},
then Wk is undefined and either detQ0(C2) = 0 (which means that the initial value WC2 at t = 0 is undefined or at least one element of the matrix W (t, WC2 ) tends to infinity as t → ∞ (in the chordal metric).
Proof. Since R(aj) ̸= R(ak) for j ̸= k, the eigenvalues λν1k,...,λνnk and the
corresponding eigenvectors in vlα(ak), v˜(ak) always appear in conjugate complex pairs if M is real. Therefore Im(vν1k,...,vνnk) and Im(v¯ν1k,...,v¯νnk) define the same invariant subspace of M, hence, from Theorem 2.1.2 and Corollary 2.1.3 we infer that Wk = Wk.
lα 1≤α≤n wk = |vν1k(l,α),...,vνnk(l,α)|,
j∈J0(C1) wlα(t;WC1)=
j∈J0(C1)
etζ(aj)v˜(aj)D(aj,C1) ,
(2.39)
lα |v˜ν1k , . . . , v˜νnk |
Let C1 be given then from (2.33)
etζ(aj )vlα(aj )D(aj , C1)
where v˜(ak)D(ak,C1) ̸= 0 and R(aj) < R(ak) for j ∈ J0(C1) \ {k}. Using (2.39) we obtain (2.38) and in the particular case that C1 = C, obviously WC1 = WC = Wk. Remark 2.4.7 also follows from (2.39) immediately if we assume v˜(ak) = 0.