 Linear flow

For a constant real square matrix $A$, the autonomous differential equation $\dot{x}=Ax$ has the linear flow ${\phi }_t(x)=e^{tA}x$, 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 $e^{t\mathrm{tr}A}$, 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.