Linear dynamical system

A linear dynamical system is a system whose state evolves according to linear equations, typically $\dot{x} = Ax$ in continuous time or {k+1} = Ax_k$ in discrete time, where $ is a constant matrix. The eigenvalues of $ determine every qualitative feature of the system's behavior: stability, oscillation frequency, and growth or decay rates. Linear dynamical systems are the foundation of control theory, state estimation, and the local analysis of nonlinear systems through linearization. Despite their simplicity, they exhibit rich phenomena including resonant modes, modal decomposition, and the interplay between spectral radius and transient growth that challenges naive Lyapunov stability analysis.