If $W$ is a continuously differentiable $n×n$ matrix valued function on the rectangle $J×J \subset \mathbf{R^2} $, how can I show that
$W(r,s)W(s,r) = I$
$W(r,s)W(s,t) = W(r,t) $
$ \forall$ $r,s,t \in \mathbf{R} $
I've got these identities to work with;
$ W'(s,t) = A(s)W(s,t) $
$\dot{W}(s,t) = - W(s,t)A(t)$
$W(s,s) = I $
$A$ is a continuous $n×n$ matrix valued function defined on $J \subset \mathbf{R} $
The above identities are generalized differential equations, look here. The solution for the $W(s,t)$ will be a path ordered exponential. There will be a function like $\Pi(r,s):=T(e^{\int_r^s A(t) dt})$ and it holds $\Pi(s,s) = I$.