stability via eigenvalues
Stability asks a simple question: if you nudge a system away from equilibrium, does it return or run away? For a linear system the answer is read straight off the eigenvalues of its matrix. Each eigenvalue is one mode; if every mode shrinks over time, the whole system is stable.
The criterion splits by time type. For a continuous system x' = A x, equilibrium is asymptotically stable exactly when every eigenvalue has negative real part, Re(lambda) < 0, since each mode carries a factor exp(lambda t). For a discrete system x_{k+1} = A x_k, stability requires every eigenvalue inside the unit circle, |lambda| < 1, since each mode carries lambda^k.
The boundary cases are delicate. An eigenvalue exactly on the line Re(lambda) = 0 (or on the unit circle |lambda| = 1) is marginal: the mode neither grows nor decays in the diagonalizable case, but if that eigenvalue is defective the Jordan block produces polynomial growth and the system is actually unstable.
Why it matters: this is the bedrock of control engineering, numerical method analysis, and judging equilibria of nonlinear systems through their linearization (the Jacobian's eigenvalues). The caveat: eigenvalues govern only long-run, asymptotic behavior. A stable non-normal system can still surge transiently before decaying, which matters in fluid turbulence and stiff numerics.
Stability is decided by the worst eigenvalue: left half-plane for flows, unit disk for steps.
Continuous vs discrete uses two different stable regions: the open left half-plane (Re(lambda) < 0) for x' = Ax, and the open unit disk (|lambda| < 1) for x_{k+1} = Ax_k. The map z = exp(lambda h) carries one into the other for a step size h.