Phasor notation of Dynamic Mode Decomposition

The paper develops the phasor-notation view of DMD and explicitly derives: (a) conjugate-pair cancellation yielding a real contribution (its Eqs. 6–8), and the real-form sum ∑ b_j e^{µ_j t}(ϕ^R_j cos ω_j t − ϕ^I_j sin ω_j t) together with conversion to a single cosine using φ = atan2(ϕ^I_j, ϕ^R_j), the practical sign convention, and the factor-of-two for a conjugate pair (its Eqs. 9–13) ; (b) S_j = |ϕ_j|, W_j = cos(ω_j t + φ_j), so x̃ = ∑ b_j S_j W_j e^{µ_j t} and, for a pair, x̃_{j,j+1} = 2 b_j S_j W_j e^{µ_j t} (Eq. 18) ; and (c) in windowed/multiscale mrCOSTS, the band summaries β_p, S_p, W_p as weighted averages with weights ∝ e^{µ t}b, including when/why this approximation can fail; it also observes that W_p lies between −1 and 1 in practice (Eqs. 25–27 and discussion) . The candidate solution matches these steps and adds two useful technical reinforcements: (i) a short argument that real-valuedness forces b_{j+1}=b_j within each conjugate pair (the paper implicitly sets equal amplitudes by writing both terms with the same b), and (ii) an exact covariance decomposition x̃_p = β_p(S_pW_p + Cov_w) with a Cauchy–Schwarz bound. These additions strengthen, and do not contradict, the paper’s claims. Overall, both are correct and follow substantially the same proof outline.