Source
Farrell & Ioannou (1996), LIT-931, read in full in NOTE-831: §2 (eqs. 1–10) and §3 (eqs. 11–19). Trefethen, Trefethen, Reddy & Driscoll (1993), LIT-932, read in full in NOTE-839: eqs. 5–13.
What was actually shown
For du/dt = Au, with A fixed in time and growth measured in an inner-product norm. Each statement is the source’s, and each was checked by its standard proof for the promotion audit.
- Normal operators follow their eigenvalues. A normal A is unitarily diagonalizable, A = UΛU†, so ‖e^(At)‖ = ‖e^(Λt)‖ = e^(α(A)t), with α the spectral abscissa (LIT-931, §2).
- Growth is a singular value. The largest growth over time t is the operator norm ‖e^(At)‖ = σ_max(e^(At)), and the optimal perturbation is the top right singular vector, by the SVD (LIT-931, eqs. 4–5).
- The eigenvalues are a floor. σ_max(e^(At)) ≥ ρ(e^(At)) = max_i |e^(λ_i t)|, since the spectral radius never exceeds the norm (eq. 6).
- Initial growth is the numerical abscissa. d/dt ‖e^(At)x‖² at t = 0 is ⟨(A + A†)x, x⟩, so the initial growth rate of ‖e^(At)‖ is λ_max((A + A†)/2) (eq. 10). It can be positive when every eigenvalue is in the left half-plane. For A = [[−1, M], [0, −2]], the symmetric part [[−1, M/2], [M/2, −2]] has trace −3 and determinant 2 − M²/4, so its top eigenvalue is (−3 + √(1 + M²))/2, positive exactly when M > 2√2.
- Forced response and pseudospectra. If A = EΛE⁻¹, then (iωI − A)⁻¹ = E(iωI − Λ)⁻¹E⁻¹, so ‖(iωI − A)⁻¹‖ ≤ κ(E)/dist(iω, Λ(A)) (LIT-931, eq. 19). The ε-pseudospectrum, the set of z with ‖(zI − A)⁻¹‖ > 1/ε, is exactly the set of eigenvalues of A + Δ with ‖Δ‖ < ε (LIT-932, eq. 13). For a non-normal A it can reach far past the ε-neighbourhood of the spectrum, so large resonance can occur far from any eigenvalue (LIT-932, eqs. 5–7).
The sources’ examples show the sizes involved. A 2 × 2 operator with eigenvalues −1 and −2 grows about eightfold before decaying. A damped Eady model, stable in every mode, grows a hundredfold in energy over four days (LIT-931, Figs. 4 and 8).
What this does not say
- Anything about white-noise variance. That a non-normal stable system holds more forced variance than the sum of its pole contributions, with equality only when A is normal, is THEORY-323. The source cites it and does not prove it.
- That non-normality belongs to the system. It is relative to the norm, and a norm can always be found that makes A normal (LIT-931, §2). The statements have content only once the norm is fixed on independent grounds, such as energy.
- That the growth persists. A stable autonomous system still decays at the rate of its spectral abscissa in the end.
- Anything for time-varying or nonlinear systems. Those are THEORY-325 and THEORY-304.