= Linear flow
{title2=$\varphi_t(x)=e^{tA}x$}
For a constant real square <matrix> $A$, the <autonomous differential equation> $\dot x=Ax$ has the <linear flow> $\varphi_t(x)=e^{tA}x$, where $e^{tA}$ is the <matrix exponential>. The identity $e^{(t+s)A}=e^{tA}e^{sA}$ gives the <flow map> composition law, and $e^{-tA}$ reverses the evolution. Its <determinant> is $e^{t\operatorname{tr}A}$, so the <matrix trace> gives the instantaneous fractional rate of <volume> change, as illustrated in <trace and local volume change>. A zero trace preserves volume even when the flow stretches one direction and contracts another.
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