Two clocks: discrete steps and continuous flow
A linear dynamical system comes in two flavors. Discrete time: x_{t+1} = A x_t, so x_t = A^t x_0 — exactly the Markov-chain iteration from guide 2, now without the probability constraint. Continuous time: dx/dt = A x, whose solution is x(t) = e^{At} x_0 using the matrix exponential. Both are governed by the eigenvalues of A.
Diagonalization decouples the system
If A = Q Lambda Q^{-1} is diagonalizable, change coordinates to the eigenbasis with y = Q^{-1} x. The coupled system becomes n independent scalar equations: dy_i/dt = lambda_i y_i, each solved by y_i(t) = e^{lambda_i t} y_i(0). The matrix exponential is then trivial: e^{At} = Q diag(e^{lambda_1 t}, ..., e^{lambda_n t}) Q^{-1}.
# Coupled springs / two tanks: dx/dt = A x with
# A = [ -2 , 1 ;
# 1 , -2 ]
# eigenvalues: lambda_1 = -1 , lambda_2 = -3 (both negative)
# eigenvectors: q1 = (1, 1)/sqrt2 , q2 = (1,-1)/sqrt2
# general solution:
# x(t) = c1 e^{-1 t} q1 + c2 e^{-3 t} q2
# both modes decay -> x(t) -> 0 . The system is STABLE.Eigenvalues alone decide stability
Since every mode behaves like e^{lambda_i t} (continuous) or lambda_i^t (discrete), stability is read straight off the spectrum. Complex eigenvalues alpha +/- i*beta add oscillation at frequency beta with envelope e^{alpha t}; it is the real part that decides growth or decay.
- Continuous dx/dt = A x: stable iff every eigenvalue has real part < 0 (left half-plane).
- Discrete x_{t+1} = A x_t: stable iff every eigenvalue has |lambda| < 1 (inside the unit circle).
- Marginal cases sit exactly on the boundary; one eigenvalue past it and the system blows up.