The complex matrix exponential of a Hermitian matrix is unitary: $e^{-iH} = U$. Is there a name or a characterization for matrices Q whose real exponential is stochastic: $e^{-Q} = S$?



Matrices whose exponential is stochastic
A matrix $A$ such that $\exp(tA)$ is (right) stochastic for all $t > 0$ should be called a "generator of a semigroup of stochastic matrices" or an "infinitesimally stochastic matrix". Clearly, since $A=\lim\_{t\to0} (\exp(tA)-I)/t$, (i) the sum of the elements in each row of $A$ has to be 0, and (ii) all non-diagonal elements must be non-negative. Conversely, a matrix $A$ satisfing (i) and (ii), for large enough $n$ produces a stochastic matrix $I+A/n$, hence $(I+A/n)^n$ and $\exp(A)=\lim\_{n\to\infty}(I+A/n)^n$ are also stochastic (and so is $\exp(tA)$). That said, I would have a look at the results of a Google search with "infinitesimally stochastic" (I can't do it now).


(Edit: as observed, the above is a stronger condition than the one you asked for; though it's a more close analog to your example.)


