Fibonacci Matrix Eigenvalues Link Golden Ratio to Aperiodic Tiling Dynamics — E8 Intelligence Research

FINDING: The Fibonacci sequence is generated by a 2×2 matrix whose eigenvalues are the golden ratio φ and its reciprocal -1/φ; the matrix powers encode substitution tiling inflation dynamics for Penrose and related aperiodic tilings. MATH: - Fibonacci matrix: \( F = {pmatrix} 1 & 1 \\ 1 & 0 {pmatrix} \), eigenvalues \( λ_1 = φ = {1+√5}{2} ≈ 1.618 \), \( λ_2 = -1/φ = {1-√5}{2} ≈ -0.618 \). - Diagonalization: \( F^n = P {pmatrix} φ^n & 0 \\ 0 & (-φ⁻¹)^n {pmatrix} P⁻¹ \), yielding Binet formula: \( F_n = {φ^n - (-φ)⁻ⁿ}{√5} \). - Substitution tiling inflation factor = φ (scale factor for Penrose kite/dart inflation). - Complex root of \( x^3 = x + 1 \) (plastic constant ≈ 1.3247) generates new substitution tilings (Ed Pegg finding). CONNECTION: - Golden ratio φ = 1.618, its reciprocal 0.618, and φ² = 2.618 appear as eigenvalues and inflation scalars. - Penrose tilin Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com