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For dx/dt = Ax with equilibrium at x = 0, stability is determined by eigenvalues: asymptotically stable if all Re(λ) < 0 (decay to origin); unstable if any Re(λ) > 0 (grow unbounded); marginally stable if Re(λ) = 0 with geometric multiplicity equal to algebraic multiplicity. Stability is geometric and visible in phase portraits, making it the lens for understanding system behavior.
From your work with phase portraits, you have seen how trajectories of x' = Ax behave geometrically: spiraling inward or outward, flowing toward or away from the origin, or orbiting around it. Stability classification systematizes these observations by connecting the geometry you saw in phase portraits directly to the eigenvalues of A — the same eigenvalues that determined the qualitative form of the solution eλt.
The fundamental rule is governed by the real parts of the eigenvalues. If all eigenvalues satisfy Re(λ) < 0, every solution decays to the origin as t → ∞, regardless of where it starts. This is asymptotic stability: the equilibrium at the origin acts as an attractor for all nearby trajectories. Physically, think of a damped oscillator — any perturbation dissipates, and the system returns to rest. If any eigenvalue has Re(λ) > 0, that mode grows exponentially, and the equilibrium is unstable: trajectories starting arbitrarily close to the origin eventually escape. A saddle point is the canonical example — stable in some directions, unstable in others, making it unstable overall.
The subtle case is marginal stability: all eigenvalues are purely imaginary (Re(λ) = 0), and each has geometric multiplicity equal to algebraic multiplicity. This second condition ensures the matrix is diagonalizable over ℂ, so no polynomial factors like teiωt appear in the solution — only pure oscillatory terms eiωt. The center equilibrium of an undamped harmonic oscillator is the canonical example: solutions orbit forever without growing or shrinking. If the multiplicities fail to match (a defective matrix), the solution contains factors like teλt, which grow even when Re(λ) = 0, making the equilibrium unstable despite purely imaginary eigenvalues.
For 2×2 systems, the classification condenses into a concrete decision tree using the trace tr(A) = λ₁ + λ₂ and determinant det(A) = λ₁λ₂. Plotting regions in the (tr, det) plane reveals the full taxonomy: det < 0 → saddle (unstable); det > 0 and tr < 0 → stable node or spiral; det > 0 and tr > 0 → unstable node or spiral; det > 0 and tr = 0 → center (marginally stable). The boundary curves separate these regions, and this single diagram unifies every phase portrait type you studied geometrically into one algebraic picture.