Matrix Diagonalization Derives Fibonacci Binet Formula via Eigenvalues — E8 Intelligence Research

FINDING: Matrix diagonalization yields closed-form Binet formula for Fibonacci numbers, linking linear recurrence to eigenvalues and the golden ratio. MATH: - Recurrence: \( F_{n+1} = F_n + F_{n-1} \), matrix form \( \begin{pmatrix} F_{n+1} \ F_n \end{pmatrix} = \begin{pmatrix} 1 & 1 \ 1 & 0 \end{pmatrix} \begin{pmatrix} F_n \ F_{n-1} \end{pmatrix} \). - Eigenvalues: \( \lambda_1 = \frac{1+\sqrt{5}}{2} = \phi \approx 1.618 \), \( \lambda_2 = \frac{1-\sqrt{5}}{2} = \psi \approx -0.618 \). - Closed form (Binet): \( F_n = \frac{\phi^n - \psi^n}{\sqrt{5}} \). - Diagonalization: \( A = PDP^{-1} \), \( D = \text{diag}(\phi, \psi) \), \( P = \begin{pmatrix} \phi & \psi \ 1 & 1 \end{pmatrix} \). CONNECTION: - Eigenvalues are golden ratio \(\phi\) and its negative reciprocal \(-\phi^{-1} = \psi\). - Ratio \( \phi = 1.618 \), \( \phi^{-1} = 0.618 \), \( \phi^2 = 2.618 \). - No direct base-60 or crystallographic symmetry, but \(\phi\) appears in quasicrystal diffraction patt Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com