Reduction of the RPA eigenvalue problem and a generalized Cholesky decomposition for real-symmetric matrices

The particular symmetry of the random-phase-approximation (RPA) matrix has been utilized in the past to reduce the RPA eigenvalue problem into a symmetric-matrix problem of half the dimension. The condition of positive-definiteness of at least one of the matrices has been imposed (where A and B are the sub-matrices of the RPA matrix) so that, e.g., its square root can be found by Cholesky decomposition. In this work, alternative methods are pointed out to reduce the RPA problem to a real (not symmetric, in general) problem of half the dimension, with the condition of positive-definiteness relaxed. One of the methods relies on a generalized Cholesky decomposition, valid for non-singular real symmetric matrices. The algorithm is described and a corresponding routine in C is given. (authors)