Page 32 - Textos de Matemática Vol. 36

22 Chapter 2. Global aspects where aij(t) denotes the minor of Qji(t) with respect to Q(t), we obtain for
−1 detQ(t)(PQ (t))lα =
detQ(t)̸=0(aswith(2.18))and1≤l≤m, 1≤α≤n, m+n
m+n
= eλktv
k,n+l
m+n m+n ...
k=1
=
e
|vν1(l,α),...,vνn(l,α)| . ,
=
m+n
λkt
k=1
α−1,n ck,n

a1α λkt  . 
×vν11 . . . vνα−1,α−1vνα+1,α+1 . . . vνnn ck1 ...
cνα+1,n .

(λν1 +...+λνn )t
cνn,1 ...cνn,n
c
ν1 .
e vk,n+l(ck1,...,ckn) .  anα
k=1
Q1,1(t) ... Q1,n(t)
. . ..
Q (t) ... α−1,1
vk,n+l ck,1 ... Qα+1,1(t) ...
. .
Qn,1(t)
m+n m+n
... e(λν1+...+λνα−1+λνα+1+...+λνn)t ν1=1 να−1=1 να+1=1 νn=1
cν1,1 ... cν1,n
. . ..
e

Before we derive general properties of the phase portrait of non-symmetric Riccati differential equations, we shall study some simple examples. These ex- amples will automatically lead us to consider a natural extension of the flow, defined by RDE in Cm×n, to a flow on a Graßmann manifold. It will also bring a geometric aspect into our considerations.
At least since the time of Poincar´e it has been known that the behavior of certain differential equations in the plane could be clarified by extending the domain to the projective plane. It was observed in [Schn73] that the natural compactification of the domain Cm×n (or Rm×n) for RDE is the Graßmann manifold Gn(Cm+n) (or Gn(Rm+n)). This viewpoint is nicely worked out in [Shay86] and [Shay91] where one can also find additional references.
1≤ν1<ν2<···<νn≤m+n
which together with (2.18) yields the desired representation formula (2.14).
Q (t)
Qα+1,n (t) .
. ... Qn,n(t)
cνα−1,1 ...
cνα−1,n ckn
cνα+1,1 ... .
.
.
c
νn