Let $$ M = \begin{pmatrix}A& B\\C& D\end{pmatrix}. $$ Is there a general formula for $e^M$ in terms of the sub-matrices $A$, $B$, $C$ and $D$? If not, can anything useful be said about the properties of $e^M$, given properties of its component blocks?
I've tried a few different approaches but haven't got anywhere, mostly because the formula $e^{A + B}=e^Ae^B$ only works if $A$ and $B$ commute.
If it helps, I'm most interested in the case where $M$ is a symmetryic real matrix, and an 'infinitesimal stochastic' matrix in particular. (An infinitesimal stochastic matrix is a symmetric real matrix whose column sums are all zero, and which has no negative off-diagonal elements.) If there are other special sub-classes of matrices for which something useful can be said here, I would be interested to know that.