Sherman–Morrison formula

In linear algebra, the Sherman–Morrison formula, named after Jack Sherman and Winifred J. Morrison, computes the inverse of a "rank-1 update" to a matrix whose inverse has previously been computed. That is, given an invertible matrix ${\displaystyle A}$ and the outer product ${\displaystyle uv^{\textsf {T}}}$ of vectors ${\displaystyle u}$ and ${\displaystyle v,}$ the formula cheaply computes an updated matrix inverse ${\textstyle \left(A+uv^{\textsf {T}}\right){\vphantom {)}}^{\!-1}.}$

The Sherman–Morrison formula is a special case of the Woodbury formula. Though named after Sherman and Morrison, it appeared already in earlier publications.