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: \( F = \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix} \), eigenvalues \( \lambda_1 = \phi = \frac{1+\sqrt{5}}{2} \approx 1.618 \), \( \lambda_2 = -\frac{1}{\phi} = \frac{1-\sqrt{5}}{2} \approx -0.618 \). - Diagonalization: \( F^n = P \begin{pmatrix} \phi^n & 0 \\ 0 & (-\phi^{-1})^n \end{pmatrix} P^{-1} \), yielding Binet formula: \( F_n = \frac{\phi^n - (-\phi)^{-n}}{\sqrt{5}} \). - 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