Let $A(t)$ be a smooth function from $[-1,1]$ to the $n \times n$ complex matrices. Define the time ordered exponential
$$\prod_{-1}^1 \exp(A(t) dt)$$
as in [this question][1], as the limit of Riemann products $\prod_{i=1}^n \exp(f(t^{\ast}_i) \ \delta t_i)$. 

The actual quantity I am interested in is
$$B(r) = \prod_{-1}^1 \exp(r A(t) dt)$$
as $r \to \infty$.

As $r \to 0$, there is a known power series expansion for $B(r)$ called the [Magnus series][2]. As $r \to \infty$, I would expect there to be something like the [stationary phase approximation][3], but I haven't been able to find it or figure it out.

I should mention that in my situation, $A(t)$ obeys
$$A(-t) = A(t)^{\ast} \quad (\dagger)$$
where $\ast$ is conjugate transpose. Condition $(\dagger)$ implies that $B(r)$ is Hermitian. I don't know whether this is helpful in any way.

  [1]: https://mathoverflow.net/questions/32705/what-is-the-standard-notation-for-a-multiplicative-integral
  [2]: http://en.wikipedia.org/wiki/Magnus_series
  [3]: http://en.wikipedia.org/wiki/Stationary_phase_approximation