A Dynamical Approach to Efficient Eigenvalue Estimation in General Multiagent Networks

The paper proves that the recoverable eigenvalues (as defined via the total weights in the Jordan expansion) are exactly the roots of a Hankel-derived polynomial p_G, with multiplicities given by m̃_i, and that rank(H) equals the sum of these m̃_i. This is stated and proved in Theorem 4 and its appendix, together with Lemma 3 on the rank of the Hankel matrix and the expansion y[k] via Jordan powers and binomial coefficients , relying on the explicit J_i^k formula and the definitions of ω_i^{(s)}, ω̄_i^{(s)}, SG, and m̃_i . The candidate model’s solution reaches the same conclusions but via a different route: it uses a shift-operator/Taylor expansion argument with basis sequences φ_{λ,s}[k] = C(k,s) λ^{k-s}, a lowering identity (S−λ)φ_{λ,s} = φ_{λ,s−1}, and an eigenvalue-isolating triangular system to force p_G^{(t)}(λ_i) = 0 for t < m̃_i. It also derives a Hankel factorization and the bound rank(H) ≤ ∑ m̃_i, then matches degrees to conclude equality and exact multiplicities. This aligns in substance with the paper’s results, though the techniques differ. A minor note: the paper’s presentation tacitly treats the displayed r×r Hankel block as invertible when rank(H) = r, which typically holds but could benefit from a brief justification or a remark about choosing an appropriate full-rank r×r minor; the model’s proof avoids committing to that specific block by arguing via annihilators and operator calculus. Overall, both are correct; proofs differ in style (Jordan-block/Hankel linear prediction vs. shift-operator/Taylor).