The system \[ \dot x=Ax+g(t),\qquad x(t_0)=x_0, \] is solved by the matrix exponential. Its spectrum and Jordan structure describe growth, oscillation, resonance, and stability.
The matrix exponential
Definition. \[ e^{tA}=\sum_{k=0}^{\infty}\frac{t^kA^k}{k!}. \] The series converges absolutely in every matrix norm, uniformly for \(t\) on compact intervals.
Proposition. \[ \frac{\mathrm{d}}{\mathrm{d}t}e^{tA} =Ae^{tA}=e^{tA}A,\qquad e^{(t+s)A}=e^{tA}e^{sA},\qquad (e^{tA})^{-1}=e^{-tA}. \]
Proof. Uniform convergence permits termwise differentiation. The product law follows by multiplying absolutely convergent series and using the binomial theorem: \[ \sum_{k+\ell=n}\frac{t^ks^\ell}{k!\ell!} =\frac{(t+s)^n}{n!}. \] The inverse identity is the product law at \(s=-t\).
Variation of constants
Theorem. \[ x(t) =e^{(t-t_0)A}x_0 +\int_{t_0}^te^{(t-s)A}g(s)\,\mathrm{d}s. \]
Proof. Put \(y(t)=e^{-tA}x(t)\). Then \[ \dot y(t)=e^{-tA}g(t). \] Integrate and multiply by \(e^{tA}\).
Jordan blocks and polynomial factors
If \(A=SJS^{-1}\), then \(e^{tA}=Se^{tJ}S^{-1}\). For a Jordan block \(J_\lambda=\lambda I+N\), \(N^m=0\), \[ e^{tJ_\lambda} =e^{\lambda t} \sum_{k=0}^{m-1}\frac{t^kN^k}{k!}. \] Thus eigenvalues determine exponential rates, while nontrivial Jordan blocks add polynomial growth. A purely imaginary eigenvalue with a Jordan block larger than one is unstable despite having zero real part.
Exponential stability
Theorem. The origin of \(\dot x=Ax\) is exponentially stable if and only if \[ \max_{\lambda\in\sigma(A)}\operatorname{Re}\lambda<0. \] Equivalently, there exist \(M,\alpha>0\) such that \[ \|e^{tA}\|\le Me^{-\alpha t},\qquad t\ge0. \]
Proof. If all real parts are negative, the Jordan formula bounds each polynomial-exponential term by \(C e^{-\alpha t}\) for any \(\alpha\) below the spectral gap. Conversely, apply the exponential bound to a complex eigenvector: \(e^{tA}v=e^{\lambda t}v\), forcing \(\operatorname{Re}\lambda\le-\alpha\).
Resonance
Example. For \[ \dot z=i\omega z+e^{i\omega t}, \] variation of constants gives \[ z(t)=e^{i\omega t}\{z(0)+t\}. \] Forcing at a natural frequency creates linear growth. In a damped system with negative spectral real part, bounded periodic forcing yields a unique bounded periodic steady response.
Nonnormal transient growth
Even when every eigenvalue is stable, \(\|e^{tA}\|\) can initially grow if \(A\) is nonnormal. The logarithmic norm \[ \mu_2(A)=\lambda_{\max}\left(\frac{A+A^\top}{2}\right) \] gives \[ \|e^{tA}\|_2\le e^{t\mu_2(A)}. \] Pseudospectra and resolvent norms describe transient sensitivity more reliably than eigenvalues alone.