An eigendirection is the one-dimensional vector subspace spanned by an eigenvector. Multiplying the eigenvector by a nonzero scalar leaves this direction unchanged. In a planar linear dynamical system, a negative real eigenvalue gives a direction of exponential approach to the equilibrium point, while a positive one gives exponential departure. For a hyperbolic saddle equilibrium, the nonlinear stable manifold and unstable manifold are tangent to the corresponding eigendirections.
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