Page 27 - Textos de Matemática Vol. 36

2.3. A representation formula
Q Ontheotherhandif P
17 isasolutionofL,thenwithW :=PQ we
−1
To prove iii) in Theorem 2.2.1 we have to observe that if we are given a
complex solution W in i) this solution is a holomorphic matrix–valued function
in G, since, as constant matrices the coefficients Mij , 1 ≤ i, j ≤ 2, are holomor-
phic. Then also Q is holomorphic in G and by the same calculations as above Q
we infer that P is a holomorphic in G solution of L.
Q Q
On the other hand if P is a solution of L then P is holomorphic
in C and by the same calculations as above we obtain that W = PQ−1 is meromorphic in C. The poles of W appear at most at points where det Q(t) = 0. Notice that in general the solutions of a matrix Riccati differential equation are always meromorphic in C if the coefficients Mij , 1 ≤ i, j ≤ 2, are holomorphic in C. But, if detQ(t) ̸= 0 for t ∈ G, then W(t) = P(t)Q−1(t) is holomorphic in G.
2.3 A representation formula
In this section we shall work out the general representation of solutions as given in (2.9). Since we want an explicit representation formula we have to stick to situations where the system L allows an explicit representation of its solutions. This, for instance, is the case if
M11 M12 (m+n)×(m+n) M=MM∈C
21 22
is a constant matrix which we shall now assume throughout this subsection. It is not our intention to present a comprehensive exposition of everything known presently on the subject. We shall make generic assumptions and focus on the properties of the differential equation under these simplifying assumptions. The general case is described in [FrJa95], [FrJa91].
Since most of the properties of RDE can be studied in this special case, we therefore assume here that M is a diagonalizable matrix, i.e., there exists an (m + n)-dimensional basis of eigenvectors
v1,...,vm+n ∈ Cm+n.
Without loss of generality we also assume that the corresponding eigenvalues
λ1, . . . , λm+n are, with respect to their real parts, ordered as
R(λ1) ≤ R(λ2) ≤ · · · ≤ R(λm+n). (2.10)
obtain
W˙ =P˙Q−1 −PQ−1Q˙Q−1 =M21 +M22W −W(M11 +M12W),
hence, W is a solution of the matrix Riccati differential equation.