Two methods are developed for computing the generalized eigenvalues and eigenvectors associated with the matrix pair {A, B}, where B is a singular Hermitian matrix and A is a matrix, not necessarily Hermitian, the range space of which is a subset of the range space of B. Conventional methods of solution break down for this case. An application is described based on the recently reported ESPRIT algorithm, a method of high-resolution direction-of-arrival estimation.