Positive matrix semigroup ($e^{tA}\ge 0(t\ge 0)$)

A matrix exponential defines a positive matrix semigroup if $e^{tA}$ maps nonnegative vectors to nonnegative vectors for all $t\ge 0$. Here the inequality means entrywise nonnegativity, with zero entries allowed. In finite dimensions this holds exactly when the real matrix $A$ has nonnegative off-diagonal entries: if $A+\gamma I\ge 0$, expand $e^{tA}=e^{-\gamma t}{\sum }_{k\ge 0}{t}^k(A+\gamma I{)}^k/k!$; conversely, the off-diagonal entries of $e^{tA}=I+tA+O(t^2)$ force $a_{ij}\ge 0$ for $i\ne j$. Zero column sums additionally preserve $1^{T}c$ under $c^{\prime }=Ac$.