Page 29 - Textos de Matemática Vol. 36

2.3. A representation formula 19
wlα(t;WC)=
1≤k1<...<kn≤m+n
c

kn

,
1≤k1<...<kn≤m+n 1≤l≤m, 1≤α≤n.

e
(λk1 +...+λkn )t
c k1
. |vk1(l,α),...,vkn(l,α)| .
 .
ν . ν  ν,n+l
vν,n
v
vν,n
and obtain the fundamental representation theorem (see [FrJa95]):
Theorem 2.3.1. (Representation formula for the solutions of RDE).
m×n −1 In i)LetWC∈C , C=V W .
C
If Q(t),P(t) are given as in (2.12), then
W (·; WC ) : C \ {t ∈ C| det Q(t) = 0} → Cm×n ,
t → P(t)Q−1(t) = W(t;WC) = (wlα(t;WC)) is a solution of RDE with W(0;WC) = WC and where
 vν,1   . 
vν,1 vν,j−1
. v˜:=.,v(l,j):= v (2.13)
 ν,j+1  . 

e
(λk1 +...+λkn )t
c
k1
. |v˜k1,...,v˜kn| .
c kn
(2.14) ii) The same solution W (·; WC ) is obtained using in i) instead of Y a solution
˜ ˜ Q0 −1
Y ofLwithinitialvalueY(0)= P withdetQ0 ̸=0andP0Q0 =WC.
0
Remark 2.3.2. i) Obviously any solution W(·;WC) defines a meromorphic matrix function with poles at most in the zeros of detQ(t), hence it is continuous (with respect to the chordal metric (see [Ahlf66])) in the com- plex plane. With respect to part ii) of Theorem 2.3.1 W(·;WC) is called the solution corresponding to Im(V C), the column space of V C.
Notice that any matrix P
0
∈ Im(V C) yields a solution Y (t) = P˜(t)
Q 0 ˜ Q˜ ( t )