= Positive matrix semigroup
{title2=$e^{tA}\ge0\quad(t\ge0)$}
A <matrix exponential> defines a <positive matrix semigroup> if $e^{tA}$ maps nonnegative <vectors> to nonnegative <vectors> for all $t\ge0$. 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 $A+\gamma I\ge0$, expand $e^{tA}=e^{-\gamma t}\sum_{k\ge0}t^k(A+\gamma I)^k/k!$; conversely, the off-diagonal entries of $e^{tA}=I+tA+O(t^2)$ force $a_{ij}\ge0$ for $i\ne j$. Zero column sums additionally preserve $\mathbf1^Tc$ under $c'=Ac$.
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