Source
Farrell & Ioannou, “Generalized Stability Theory. Part I: Autonomous Operators” (1996), LIT-931, read in full in NOTE-831: §2 (eqs. 1–10), §3 (eqs. 11–31), §4 (examples). Trefethen, Trefethen, Reddy & Driscoll, “Hydrodynamic Stability Without Eigenvalues” (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 a chosen inner-product norm:
- Normal operators are governed by their eigenvalues. If AA† = A†A, ‖e^{At}‖ = e^{λ^R_max t}, with λ^R_max the largest real part of an eigenvalue (LIT-931, §2).
- Non-normal operators are not, except as t → ∞.
- Growth over time t is ‖e^{At}‖, the largest singular value of the propagator, and the optimal perturbation is the matching right singular vector (LIT-931, eqs. 4–5).
- The eigenvalues give a floor, not a ceiling: σ²_max ≥ e^{(λ_i+λ_i*)t} for every eigenvalue (eq. 6).
- At t → 0 the growth rate is the numerical abscissa, the top eigenvalue of (A + A†)/2, which can be positive when every mode decays (eq. 10).
- At t → ∞ the leading mode wins, but it is best excited by its adjoint vector, not by itself (eqs. 8–9).
- The forced response is larger too.
- The response to harmonic forcing is bounded by κ(E)/dist(iω, Λ(A)), not 1/dist (LIT-931, eq. 19).
- Large resonance can occur far from any eigenvalue: “pseudoresonance” (LIT-932, eqs. 5–7).
- Under spatially uncorrelated white noise, the spectral density trace(R†R) is at least the sum of the pole contributions, “with equality only when A is normal” (LIT-931, eq. 22, cited to Ioannou 1995). A non-normal system holds more variance than the normal system with the same spectrum.
- The spectrum is fragile. An operator perturbation of norm ε can move eigenvalues anywhere in the ε-pseudospectrum, which for a non-normal operator may reach far past the ε-neighbourhood of the spectrum (LIT-932, eq. 13).
- Sizes in examples.
- A 2 × 2 operator with eigenvalues −1 and −2 and eigenvectors at angle π/100 grows about eightfold before decaying.
- A damped Eady model, stable in every mode, grows a hundredfold in energy over four days.
- A damped channel with only decaying modes grows its two-day optimal 112-fold (LIT-931, Figs. 4, 8, 12).
What could have come out otherwise: the examples. The general statements are theorems of linear algebra, quoted with their standard proofs or cited.
What this does not say
- It does not say the growth persists. In a stable autonomous system every perturbation still decays in the end, at the rate of the spectral abscissa. The claim is about the finite-time and forced behaviour, not the asymptote.
- It does not say non-normality is a property of the system alone. It is relative to the norm. “It is always possible to find a (usually unphysical) norm that renders the operator normal” (LIT-931, §2). The claim has content only once a norm is fixed on independent grounds, such as energy.
- It does not say what happens nonlinearly. That transient growth hands disturbances to nonlinear dynamics and so causes transition to turbulence is a separate claim. In LIT-932 it is a conjecture (THEORY-304).
- It does not say time-varying operators settle. For operators that change in time, non-normality can produce asymptotic instability, not only transient growth (THEORY-325).
- It does not say directed coupling is the source of non-normality. The sources in these papers are background shear. A one-way coupling such as the upper-triangular 2 × 2 example is non-normal, but neither paper states a general rule about directed structures.
Connections
No relations are declared.
- CLAIM-186. Its note “settling does not rule out bursts” cites these papers for exactly this claim, and states it at the right strength. What this account adds is the forced case: under continual small disturbance a non-normal system shows more standing variance than its spectrum predicts, not only occasional bursts. And it adds the caveat that the reading across needs a norm.
- THEORY-304. The shear-flow case, with scalings in Reynolds number.
- THEORY-325. The time-dependent case, where transient growth can become asymptotic instability.
Note of 2026-10-19: what moved, and what stays
The finite-time and resolvent statements in “What was actually shown” are now THEORY-727, Active, with their textbook proofs written out. They cover normal operators, growth as a singular value, the eigenvalue floor, the numerical abscissa, the harmonic-forcing bound and pseudospectra. What this THEORY still holds is the white-noise clause: trace(P) ≥ Σ 1/(−2 Re λ_i), with equality only for normal A. LIT-931 cites it and the record has not read the proof, so it stays Proposed.