An N^(3/2)logN^(3/2) Method for Applying the Hankel Eigenvector Matrix

Applying the Hankel eigenvector matrix to a vector is the projection primitive ofsubspace spectral estimation and structured deconvolution. Forming that matrixcosts O(N³) arithmetic and O(N²) storage, and each later application costs O(N²).We show that the expense is not inherent. For a real symmetric Hankel matrix the transpose of the eigenvector matrix factorsexactly as Vᵀ = W⁻ᵀKᵀ, with K block Krylov in A and W block Vandermonde in theeigenvalues; a block companion relation reduces the resulting Vandermonde solve toone polynomial product plus one multipoint evaluation; and the eigenbasisrepresentation of the downshift obeys a displacement equation of rank at mostfour, making it a Cauchy-type matrix generated by four vectors read off the Hankelentries. Balancing the block width at s ≈ √(N/log²N) yields a query cost ofΘ(N^(3/2) log^(3/2) N) against Θ(N²) dense, a reduction by a factorN^(1/2)/log^(3/2) N; the identities hold to rounding, 10⁻¹⁴. The second half delimits when the bound is attainable, and it is not attainable atany size computable in double precision. Calibration must materialise V, at Θ(N²)storage, which is 80 GB at N = 10⁵ and 8 TB at N = 10⁶. And a direct realizationof the second stage is limited by cond(W), which reaches 1.2×10¹⁹ at width 32 evenafter the blocks are scaled: the relative error of a query is already 5×10¹ atwidth 2 and 2×10⁴⁹ at width 32. The bound is therefore a statement aboutarithmetic. It becomes favourable only above N ≈ 10⁴, where the calibration itassumes is already out of reach, and we regard the location of that boundary,rather than a usable algorithm, as what this paper delivers.