A Jacobi Algorithm in Phase Space: Diagonalizing (skew-) Hamiltonian and Symplectic Matrices with Dirac-Majorana Matrices

TL;DR

This paper extends Jacobi's diagonalization method to Hamiltonian phase spaces, enabling efficient eigenvalue and eigenvector computation for Hamiltonian and skew-Hamiltonian matrices using symplectic transformations.

Contribution

It introduces a generalized Jacobi algorithm that operates in phase space with symplectic transformations for diagonalizing Hamiltonian matrices.

Findings

Successfully diagonalizes Hamiltonian matrices with real eigenvalues

Handles skew-Hamiltonian matrices with imaginary eigenvalues

Provides a symplectic analogue to classical Jacobi rotations

Abstract

Jacobi's method is a well-known algorithm in linear algebra to diagonalize symmetric matrices by successive elementary rotations. We report about the generalization of these elementary rotations towards canonical transformations acting in Hamiltonian phase spaces. This generalization allows to use Jacobi's method in order to compute eigenvalues and eigenvectors of Hamiltonian (and skew-Hamiltonian) matrices with either purely real or purely imaginary eigenvalues by successive elementary symplectic "decoupling"-transformations.