[AntonioLao]

Eigenvalues or characteristic values (l) are the roots of the characteristic equation det(A-lI)=0 where A is a square matrix which can either singular or nonsingular and I is the nonsingular self inverse identity matrix with determinant (det) unity.

If A are 2 by 2 symmetric Hadamard matrices then the two eigenvalues are 0 and 2 corresponding to the 4 linearly independent eigenvectors: (1, 1), (-1, 1), (1, -1), and (-1, -1). In a matter dominated universe only (1, 1) and (1, -1) or (-1, 1) are sufficient since (-1, -1) and (-1, 1) or (1, -1) represent the eigenvectors for antimatter. In Dirac theory these are equivalent to the 4 2-component spinors with the distinction that these are imaginary numbers for describing the abstract 720° rotation symmetry instead of 360° real symmetries. The advantage is describing spin-½ fermions while spin-0 is properly described by Klein-Gordon-Fock-Schrödinger theory in relativistic quantum mechanics for mesons and other spin-0 bosons.

Recently, Berry and Keating from the University of Bristol used random matrices extending Hilbert-Pólya-Montgomery-Dyson conjecture that non-trivial zeros of the complex Riemann zeta function is equivalent to the physical fact (and math fact) that all eigenvalues are real as products of conjugate complex number pairs. However, finding a particular infinitely ordered matrix satisfying this requirement is none other than that of a Diophantine division with inverse additive multiples square symmetric Hadamard matrices. Some more important consequences are Montgomery-Odlyzko law, Selvam’s quantum chaos, Penrose’s tiling patterns, Brownian motion in diffusion and percolation, Xiao-Song Lin’s Jones polynomial in knot theory, and Ihara-Hashimoto-Bass zeta function in graph theory. Reference: David Wells, Prime Numbers: The Most Mysterious Figures in Math, Wiley, NJ, 2005.