Diagonalization of general 3×3 Hermitian matrices

Abstract

Diagonalization of Hermitian matrices plays an important role in quantum mechanics, neutrino oscillation studies, and computational linear algebra. In particle physics, Hermitian mass matrices arise naturally in the description of neutrino mixing phenomena, where the extraction of physical eigenvalues and mixing parameters requires stable and accurate diagonalization procedures. In the present work, a Newton–style iterative framework for the diagonalization of Hermitian 3 × 3 matrices has been developed and implemented computationally. The method reformulates the diagonalization problem as a nonlinear iterative system based on unitary transformations parameterized through rotation angles and complex phase parameters. Numerical Jacobian evaluation, least-squares based correction methods, and damping techniques were incorporated to improve convergence stability and computational robustness. The framework was implemented using Wolfram Mathematica and applied to Hermitian neutrino mass matrices constructed from experimentally measured neutrino oscillation parameters. The iterative procedure converged successfully toward diagonal form while preserving the Hermitian structure and unitarity of the transformation matrix. The obtained eigenvalues showed excellent agreement with standard built-in numerical diagonalization methods, confirming the accuracy of the developed approach. The results demonstrate that the proposed iterative framework provides a stable and computationally reliable method for the diagonalization of Hermitian matrices relevant to neutrino phenomenology and computational particle physics.