Mean Subtraction and Mode Selection in Dynamic Mode Decomposition

The paper proves that, when Z = ČΘ with Č full column rank and n ≥ r, the DMD eigenvalues with nontrivial modes coincide exactly with the Koopman eigenvalues: σ_nontriv[ČΘ, c*(ČΘ)] = σ(Λ) (Theorem 5.1), with the key mechanism supplied by Appendix D (Lemma D.2) showing that true Koopman eigenvalues yield nonzero DMD modes while spurious eigenvalues yield trivial modes . The candidate’s counterexample hinges on treating the Vandermonde power vector [1, μ, …, μ^{n−1}]^T as a right eigenvector of the companion T[c*]; in this paper’s convention, those Vandermonde rows are left eigenvectors (since (V*)^{-1} is Vandermonde), and Lemma D.2 then forbids a nonzero Koopman eigenvalue from having a trivial DMD mode, contradicting the counterexample . Thus the paper’s result stands; the model’s claimed counterexample arises from an eigenvector orientation mix-up.