Sherman-Morrison Formula

The Sherman-Morrison formula is a result in linear algebra that computes the inverse of a matrix after it has been modified by a rank-one update, without having to invert the new matrix from scratch. Given an invertible matrix A and column vectors u and v, the formula expresses the inverse of A plus the outer product of u and v in terms of the already known inverse of A, adjusted by a correction term built from u and v. This makes it far more efficient than recomputing a full matrix inverse, requiring on the order of the square of the matrix dimension in scalar multiplications rather than the cube. The formula is named for Jack Sherman and Winifred J. Morrison, who published it between 1949 and 1950, though the same identity had appeared in earlier work and was later generalized into the Woodbury matrix identity. It is used in numerical computing and in theoretical physics, including calculations involving particle propagators in quantum field theory.