Eigenvalues Classify Linear System Stability

For the linear system $\mathbf x'=A\mathbf x$, eigenpairs satisfy $A\mathbf v=\lambda\mathbf v$. Along an eigenvector, the solution changes like $e^{\lambda t}\mathbf v$.

Real parts control stability

If every eigenvalue has negative real part, the equilibrium at the origin is asymptotically stable. Any positive real part creates an unstable direction. Mixed positive and negative real eigenvalues produce a saddle.

Complex eigenvalues introduce rotation. A negative real part gives an inward spiral; a positive real part gives an outward spiral. Purely imaginary eigenvalues require care, especially for nonlinear systems.

Nonlinear systems

Near an equilibrium $\mathbf x^*$, the Jacobian $D\mathbf f(\mathbf x^*)$ supplies a linear approximation. When no eigenvalue has zero real part, its classification usually determines the local nonlinear behavior.

Classify from trace and determinant

For a $2\times2$ matrix, eigenvalues satisfy

$\lambda^2-\tau\lambda+\Delta=0,$

where $\tau=\operatorname{tr}A$ and $\Delta=\det A$. Negative determinant implies eigenvalues of opposite sign and a saddle. Positive determinant with negative trace generally indicates stability; the discriminant $\tau^2-4\Delta$ distinguishes real nodes from complex spirals.

Check your understanding

What does $\det A<0$ guarantee in two dimensions?

Show the reasoning

The eigenvalue product is negative, so the real eigenvalues have opposite signs. One direction grows and one decays: the equilibrium is a saddle.