Page 30 - Textos de Matemática Vol. 36

20
Chapter 2. Global aspects
ii)
kn
called Plu¨cker coordinates of C (or of the n-dimensional subspace Im(C)
of C(m+n)×(m+n) ); these coefficients and also the coefficients |v˜k1 , . . . , v˜kn | and |vk1(l,α),...,vkn(l,α)| are coupled as we later shall see in Lemma 2.4.2. This has strong influence on the behavior of the solutions of RDE and on the structure of the set of all solutions of ARE.
Since (2.14) represents all solutions of RDE, all solutions of ARE are also represented. If for instance for a tuple of indices (k1, . . . , kn), |v˜k1,...,v˜kn| ≠ 0 holds, then by choosing C such that all determinants with rows from C are zero except |(cTk1 , . . . , cTkn )| ≠ 0, we obtain a solution of ARE by
wlα = |vk1(l,α),...,vkn(l,α)|, (2.15) |v˜k1,...,v˜kn|
1≤l≤m, 1≤α≤n.
There are now three different possibilities for describing solutions of ARE:
• By using M-invariant subspaces, as already described in Section 2.1 (see Theorem 2.1.2).
• By the representation via formula (2.15).
• By taking limits (in chordal metric) as t → ∞ in certain sectors of the complex plane of all functions defined by (2.14). This will be described in more detail in Section 2.4.
By dealing with a Jordan basis of generalized eigenvectors of M, a simi- lar representation formula to (2.14) can be obtained, as it was shown in [FrJa95]. In particular part i) of this remark remains true.
iii)
iv)
˜ Q0In
withY(0)= P = W Q0, detQ0̸=0,suchthat
0C
W(·) = P˜(·)Q˜−1(·) isasolutionofRDEwithW(0)=P Q−1 =W .Hence,W(·)=W(·;W ),
00CC i.e., all solutions of L with initial data in Im(V C) yield the same solution
W(·;WC) of RDE. c
k1 .
. appearing in (2.14) as coefficients are the so- c
The determinants
Later we shall look in more detail at the different approaches mentioned above in iii); however, first we present the proof of the representation formula.