I'm trying to solve the following question:
Establish the existence of $N \times N$ matrices $S(x)$ and $C(x)$ such that
$S'(x)=Q(x)c(x),\\ C'(x)=-Q(x)S(x),\\ S(0)=0,$
where $Q(x)$ is a given continuous $N \times N$ matrix and $0$ and $I$ denote the zero and unit matrices. Now define
$y=CC^*+SS^*, Z=CS^*-SC^*$
where the asterisk denotes the adjoint matrix. Assuming $Q$ is symmetric, find a first order matrix system satisfied bt $Y$ and $Z$ and deduce that $Y(x)=I, Z(x)=0$. What does this result reduce when $N=1$?
Any help?