A Sort-jacobi Algorithm on Semisimple Lie Algebras

A structure preserving Sort-Jacobi algorithm for computing eigenvalues or singular values is presented. The proposed method applies to an arbitrary semisimple Lie algebra on its (−1)-eigenspace of the Cartan involution. Local quadratic convergence for arbitrary cyclic schemes is shown for the regular case. The proposed method is independent of the representation of the underlying Lie algebra and generalizes well-known normal form problems such as the symmetric, Hermitian, skew-symmetric, symmetric and skew-symmetric R-Hamiltonian eigenvalue problem and the singular value decomposition. Due to the well-known classification of semisimple Lie algebras, the approach yields structure-preserving Jacobi eigenvalue methods for so far unstudied problems as Takagi’s and a Takagi-like factorization, the Hermitian quaternion, the Hermitian C-Hamiltonian eigenvalue decomposition and a symplectic singular value decomposition. The algorithm is exemplified for an exceptional Lie algebra.