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: \( Fₙ₊₁ = F_n + Fₙ₋₁ \), matrix form \( {pmatrix} Fₙ₊₁ \\ F_n {pmatrix} = {pmatrix} 1 & 1 \\ 1 & 0 {pmatrix} {pmatrix} F_n \\ Fₙ₋₁ {pmatrix} \). - Eigenvalues: \( λ_1 = {1+√5}{2} = φ ≈ 1.618 \), \( λ_2 = {1-√5}{2} = ψ ≈ -0.618 \). - Closed form (Binet): \( F_n = {φ^n - ψ^n}{√5} \). - Diagonalization: \( A = PDP⁻¹ \), \( D = diag(φ, ψ) \), \( P = {pmatrix} φ & ψ \\ 1 & 1 {pmatrix} \). CONNECTION: - Eigenvalues are golden ratio \(φ\) and its negative reciprocal \(-φ⁻¹ = ψ\). - Ratio \( φ = 1.618 \), \( φ⁻¹ = 0.618 \), \( φ^2 = 2.618 \). - No direct base-60 or crystallographic symmetry, but \(φ\) appears in quasicrystal diffraction patt Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com