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 A {\displaystyle A} and the outer product u v T {\displaystyle uv^{\textsf {T}}} of vectors u {\displaystyle u} and v , {\displaystyle v,} the formula cheaply computes an updated matrix inverse ( A + u v T ) ) − 1 . {\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.
Statement Suppose A ∈ R n × n {\displaystyle A\in \mathbb {R} ^{n\times n}} is an invertible square matrix and u , v ∈ R n {\displaystyle u,v\in \mathbb {R} ^{n}} are column vectors. Then A + u v T {\displaystyle A+uv^{\textsf {T}}} is invertible if and only if 1 + v T A − 1 u ≠ 0 {\displaystyle 1+v^{\textsf {T}}A^{-1}u\neq 0} . In this case,
( A + u v T ) − 1 = A − 1 − A − 1 u v T A − 1 1 + v T A − 1 u . {\displaystyle \left(A+uv^{\textsf {T}}\right)^{-1}=A^{-1}-{A^{-1}uv^{\textsf {T}}A^{-1} \over 1+v^{\textsf {T}}A^{-1}u}.}
Here, u v T {\displaystyle uv^{\textsf {T}}} is the outer product of two vectors u {\displaystyle u} and v {\displaystyle v} . The general form shown here is the one published by Bartlett.
Proof ( ⇐ {\displaystyle \Leftarrow } ) To prove that the backward direction 1 + v T A − 1 u ≠ 0 ⇒ A + u v T {\displaystyle 1+v^{\textsf {T}}A^{-1}u\neq 0\Rightarrow A+uv^{\textsf {T}}} is invertible (with inverse given as above) is true, we verify the properties of the inverse. A matrix Y {\displaystyle Y} (in this case the right-hand side of the Sherman–Morrison formula) is the inverse of a matrix X {\displaystyle X} (in this case A + u v T {\displaystyle A+uv^{\textsf {T}}} ) if and only if X Y = Y X = I {\displaystyle XY=YX=I} . We first verify that the right hand side ( Y {\displaystyle Y} ) satisfies X Y = I {\displaystyle XY=I} .
… excerpt ends here. Continue reading the full article.
