A sort-Jacobi algorithm for semisimple Lie algebras
The classical Jacobi algorithm is extended to an unified Lie algebraic approach. The conventional Jacobi algorithm minimizes the distance to diagonality; they reduce the off-norm, i. e. the sum of squares of off-diagonal entries. Sorting the diagonal elements after each step would accelerate the convergence but, there are difficulties to apply this sorting to the off-norm, that has to minimize. Using the gradient flow of a trace function [\textit{R. W. Brockett}, Linear Algebra Appl. 146, 79--91 (1991; Zbl 0719.90045)] as a more appropriate distance measure matrices can be diagonalized, and the eigenvalues can be simultaneously sorted. In this paper the sort-Jacobi algorithm is extended to a large class of structured matrices, besides the semisimple Lie algebra cases, such as e. g. the symmetric, Hermitian, and skew-symmetric eigenvalue problem, and the real and complex singular value decomposition. The local quadratic convergence of the sort-Jacobi method is proved for the regular case but not for the case in which the eigenvalues or singular values occur in clusters. For these more complicated irregular problem a future publication is announced. The results are illustrated considering the Lie algebra of derivations of the complex octonions.
scientific article; zbMATH DE number 5376893
scientific article