How to compute time ordered Exponential? So say you have a matrix dependent on a variable t:
$A(t)$
How do you compute $e^{A(t)}$ ?
It seems Sylvester's formula, my standard method of computing matrix exponentials can't be applied here given the varying nature of the matrix and furthermore the fact that it may not always have distinct eigenvalues.

There seems to be some confusion here between two different problems: the first is the computation of a time-ordered (also called path-ordered) exponential, a matrix E(t',t) solving $\frac{d}{dt}E(t,0) = M(t)E(t,0) $ and the computation of the matrix exponential of a time dependent matrix $\exp(A(t))$. 
Recall that the first and second problems are related whenever $M$ commutes with itself at different times. Indeed if $M(t')M(t)-M(t)M(t')=0$ for all t,t', then $E(t)=\exp(\int_0^t M(\tau)d\tau)$ and so the time-ordered exponential is an ordinary matrix exponential. It is then straightforward to compute with the usual expm (Matlab) or MatrixExp (Mathematica) functions and the likes. 
If the matrix $M$ does not commute with itself at different times, then $E(t)$ is not an ordinary matrix exponential. As notedin Yiteng's response, the problem is then much more difficult and admits few analytical answers. As far as I can tell, since Magnus-series are impossible (baring rare exceptions?) to compute exactly to all orders, the only analytical approach is the path-sum formulation thing, which looks quite difficult. If numerical is sufficient for you, then you can always Magnus the hell out of it.
