A block Jacobi method for complex skew-symmetric matrices with applications in the time-dependent variational principle

An iterative Jacobi-like algorithm is described for transforming a skew-symmetric complex matrix A of even dimension into Murnaghan’s normal form. The decomposition allows to determine the singular values of A and to solve the system of linear equations Ax = b in a least square sense without accidentally destroying the skew-symmetry. Complex skew-symmetric matrices arise in the context of the time-dependent variational principle (TDVP), that maps a quantum mechanical system to a Hamiltonian system in a high-dimensional curved phase space. When the skew-symmetric phase space metric becomes singular because of parameter redundancies, the equations of motion have to be solved in a least square sense. The presented algorithm ensures that the symplectic structure is retained also in the least square solution. As a test case it is applied to studying the deflection of a photoelectron by a hydrogen atom using the TDVP. A Fortran implementation of the skew-Jacobi algorithm is provided as supplementary material.