Updating and Downdating Matrix Decompositions

Abstract

The Sherman–Morrison–Woodbury formula ([GWS-B7], p. 328) is a recipe for constructing the inverse of a matrix after it has been modified by a low-rank correction. For a matrix of size n ×n that has been so modified, it enables the inverse of this matrix to be updated in time proportional to kn, where k is the rank of the correction, rather than the n ³ time usually necessary to compute the inverse directly. This important fact has enabled a variety of algorithms, from early implementations of the simplex method for linear optimization [29] to algorithms for solving least squares problems when new data arrive.