Page 31 - Textos de Matemática Vol. 36

2.3. A representation formula 21 Proof. ii) follows from uniqueness of the solution of the initial value problem
for RDE. To obtain i) we use the notation in (2.13) and infer from (2.12)
Q(t) =
=
m+n m+n m+n cν1eλνtv˜ν, cν2eλνtv˜ν,..., cνneλνtv˜ν
ν=1 ν=1 ν=1 m+n 
vk1eλktck Q (t) ... Q (t)
k=1 11 1n
(2.16)
and
P(t) =

m+n 

m+n
ν=1
 vν,n+1  λνt . 
.. .
 .
m+n
v eλktc
Qn1(t) ... Qnn(t)
ν=1
= . . 

 vν,n+1  λνt . 
kn k k=1
cν1e
 . ,..., vν,m+n
cνne
 .  vν,m+n
m+n  vk,n+1eλktck
 k=1 
(2.17)
 .  = . .
m+n  vk,m+neλktck
k=1
Using the multi–linearity of the determinant we then obtain
m+n m+n
detQ(t)= ... e(λk1+...+λkn)tck11,...,cknn|v˜k1,...,v˜kn|=
k1 =1 kn =1

1≤k1 <k2 <...<kn ≤m+n
e(λk1 +...+λkn )t|v˜k1 , . . . , v˜kn |( sign (π)ckπ(1)1 · · · ckπ(n)n) π∈σn
c k1
. e 1 n |v˜k1,...,v˜kn| . .
c kn
=

1≤k1 <k2 <...<kn ≤m+n
(λk +...+λk )t
(2.18) Since det Q(t) is an exponential sum with constant coefficients, the asymptotic distribution of the zeros of det Q(t) is well known (see Lemma 1, [Boes89] and [Lang31]). In order to calculate (PQ−1)lα, the element in the l − th row and
α − th column of P Q−1 , we use (2.16)–(2.18). From m+n
and
a11(t) . . . a1n(t) −1 1. .
Qαβ(t) = ckβeλktvkα (2.19) k=1
Q (t)=detQ(t) . . , an1(t) . . . ann(t)