## Randomized methods for rank-deficient linear systems

Josef Sifuentes, Zydrunas Gimbutas, and Leslie Greengard

### Abstract

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.

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### Key words

rank-deficient systems, null space, null vectors, eigenvectors, randomized algorithms, integral equations

### AMS subject classifications

15A03, 15A12, 15A18, 65F15, 65F99

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