Page 28 - Textos de Matemática Vol. 36

18 Chapter 2. Then if we take the non–singular matrix
Global aspects
it follows that
V =(v1,...,vm+n)∈C(m+n)×(m+n) V−1MV =diag(λ1,...,λm+n)=:∆
holds. If W0 ∈ Cm×n is given, then we define cc...c
(2.11)
(m+n)×n
or V C = W
Y (0) = W
C
1 1,1 1,n
 .   . C= . = .
.  −1 In
. :=V W ∈C
0
,
cm+n cm+n,1 In
. . .
. Notice that C(= C(W0)) is uniquely determined by W0 and
cm+n,n
0
vice versa, if we are given C ∈ C(m+n)×n, then also W0 is uniquely defined. Therefore, if we want to emphasize this dependence of W0 on the parameter matrixCweshalluseWC insteadofW0.Thenthe(m+n)×nmatrixsolution Y of the linear system L with
(2.12)
In
( see Theorem 2.2.1 ) has the form
∆t −1 In ∆t Q(t) Y(t)=Ve V W =Ve C= P(t) ,
C
t ∈ R (or t ∈ C), with Q(t) ∈ Cn×n,P(t) ∈ Cm×n and clearly Y(0) = VC. Hence, if we choose either W0 or the parameter matrix C as defined above, such that W0 = WC, then from (2.9) and (2.12) we infer that W(t) = P(t)Q−1(t) represents a solution of RDE at least in a small time interval starting at t0, or in a sufficient small neighborhood of t0 if we consider a complex t-parameter.
For systemic reasons we sometimes shall allow t to be a complex variable which does not cause any additional difficulties in the following considerations.
Let
and for 1 ≤ j ≤ n and 1 ≤ l ≤ m, we then set
 vν,1  .
vν = . , 1≤ν≤m+n, vν,m+n