Properties of matrix exponential I know that the solution to system $x' = Ax$ is $e^{At}$, and I'm aware of various methods to calculate the exponential numerically. However I wonder if there are some analytical results.
Namely, I'm interested in matrices of type $A_{i,i} = -A_{i+1,i}$; $A_{n,n}=0$. These matrices are the infinitesimal generators of a Markov chain, where you transit from state 1 to state n through various steps. Would it be possible to calculate $[e^{At}]_{n,1}$, i.e. transition probability from state $1$ to state $n$ analytically as a function of time $t$?

So all you are saying is that it's in fact a mixture of exponential distributions!? Quite amazning, if true. Can you give some references?
Igor: they're zero. The last replier has the correct interpretation of the problem, but I'm not certain yet, if the answer is right.
