Data-driven resolvent analysis

The paper develops resolvent analysis in a Q-weighted inner product and shows that gains and modes come from the SVD of the weighted resolvent FH(ω)F^{-1} (equations (3)–(5)) and, when projected onto an invariant r-dimensional eigen-subspace with AV_r = V_rΛ_r and W_r^* Q V_r = I, from the SVD of the reduced, weighted operator F̃(−iωI − Λ_r)^{-1}F̃^{-1}; the physical modes are synthesized as Φ = V_rF̃^{-1}Φ̃ and Ψ = V_rF̃^{-1}Ψ̃ (equations (9)–(13)) . The candidate solution reproduces exactly these steps: it (i) uses (−iωI − A)V_r = V_r(−iωI_r − Λ_r) to obtain H(ω)V_r = V_r(−iωI_r − Λ_r)^{-1}, (ii) introduces the reduced Q-weight F̃ via V_r^*QV_r = F̃^*F̃, and (iii) identifies the optimal restricted gain and modes from the SVD of F̃(−iωI_r − Λ_r)^{-1}F̃^{-1}, with lifting via V_rF̃^{-1}. It also explicitly states the invertibility condition −iω ∉ spec(Λ_r), consistent with the paper’s setup. Hence both are correct and follow the same proof strategy, with the model adding a clear Q-orthonormality check that is consistent with the paper’s constructions .