linear dynamical system

A linear dynamical system describes how a state evolves when the rule for change is a single matrix applied over and over. Whether time ticks discretely or flows continuously, the entire future is determined by the state now and the matrix A, and the eigenvalues of A decide whether things grow, shrink, spin, or settle.

Two flavors: the discrete system x_{k+1} = A x_k, whose solution is x_k = A^k x_0, and the continuous system x' = A x, whose solution is x(t) = exp(At) x_0. In both, diagonalizing A is the key trick: writing x_0 in an eigenbasis decouples the system into independent scalar modes, each evolving by its own eigenvalue.

In eigencoordinates a discrete mode scales by lambda^k each step and a continuous mode by exp(lambda t). So a real eigenvalue gives growth or decay, a complex pair gives rotation (oscillation), and the magnitudes set the rates. The matrix exponential exp(At) is just this eigenmode picture reassembled, the continuous-time analog of repeated multiplication.

Why it matters: linear systems model springs, circuits, populations, control loops, and the local behavior of any nonlinear system near an equilibrium (via linearization). The caveat: if A is defective (not diagonalizable), pure eigenmodes are not enough and you get polynomial-times-exponential terms from the Jordan form, which can transiently grow even in a stable system.