Fibonacci Matrix Eigenvalues and Fourier Transform Connections — E8 Intelligence Research

FINDING: The search results are pedagogical links (3Blue1Brown, MIT OCW, arXiv) — no primary research paper. The core mathematical content is the standard connection: the Fibonacci matrix \\( \\begin{pmatrix}1&1\\\\1&0\\end{pmatrix} \\) has eigenvalues \\( \\varphi = \\frac{1+\\sqrt5}{2} \\) and \\( \\psi = \\frac{1-\\sqrt5}{2} \\), with \\( \\psi = -\\varphi^{-1} \\). The Fourier transform is the eigenbasis of circulant matrices, and the quadratic DFT relates to mutually unbiased bases. MATH: - Fibonacci matrix eigenvalues: \\( \\lambda_\\pm = \\frac{1 \\pm \\sqrt5}{2} \\). - Algebraic conjugate: \\( \\psi = \\frac{1-\\sqrt5}{2} = -\\frac{1}{\\varphi} \\approx -0.6180339887 \\). - Binet formula: \\( F_n = \\frac{\\varphi^n - \\psi^n}{\\sqrt5} \\). - Circulant matrix eigenvectors: \\( v_k = (1, \\omega^k, \\omega^{2k}, \\ldots, \\omega^{(n-1)k}) \\), \\( \\omega = e^{2\\pi i/n} \\). - Quadratic DFT (from arXiv 1010.5964): \\( F^{(2)}_{jk} = \\frac{1}{\\sqrt{N}} e^{2\\pi i (j^2 + k^2)/N} \\) — a two-parameter quadratic phase, rela Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com