The Matrix Exponential as a Flow

The linear system $\dot{\mathbf{x}} = A\mathbf{x}$ has one clean solution, $\mathbf{x}(t) = e^{At}\mathbf{x}_0$, and the matrix exponential is the operator that slides every starting point forward in time at once, the flow of the system. What the flow looks like is decided entirely by the eigenvalues of $A$. Two real eigenvalues of the same sign make a node, points racing straight out from the origin or straight in; opposite signs make a saddle, drawn in along one eigendirection and flung out along the other; a complex pair makes a spiral, winding in if the real part is negative and out if it is positive; and a purely imaginary pair makes a centre, closed loops that neither grow nor decay. The scene draws the phase portrait with streamlines and a cloud of markers actually flowing along $e^{At}$, the orange dashed lines marking the real eigenvector directions the flow cannot leave. Drag the green initial point and watch the white dot ride its trajectory, the exact $e^{At}\mathbf{x}_0$. The bottom panel is the deciding diagram: the two eigenvalues plotted in the complex plane, stable to the left of the imaginary axis and unstable to the right, real on the axis and complex off it. Cycle the gallery and watch a small move of the eigenvalues across a boundary flip the whole character of the flow.