Randomized methods for rank-deficient linear systems

We present a simple, accurate method for solving consistent, rank-deficient linear systems, with or without additional rank-completing constraints. Such problems arise in a variety of applications such as the computation of the eigenvectors of a matrix corresponding to a known eigenvalue. The method is based on elementary linear algebra combined with the observation that if the matrix is rank-k deficient, then a random rank-k perturbation yields a nonsingular matrix with probability close to 1.