Woodbury matrix identity

In mathematics (specifically linear algebra), the Woodbury matrix identity, named after Max A. Woodbury,[1][2] says that the inverse of a rank-k correction of some matrix can be computed by doing a rank-k correction to the inverse of the original matrix. Alternative names for this formula are the matrix inversion lemma, Sherman–Morrison–Woodbury formula or just Woodbury formula. However, the identity appeared in several papers before the Woodbury report.[3][4]

The Woodbury matrix identity is[5]

${\displaystyle \left(A+UCV\right)^{-1}=A^{-1}-A^{-1}U\left(C^{-1}+VA^{-1}U\right)^{-1}VA^{-1},}$

where A, U, C and V are conformable matrices: A is n×n, C is k×k, U is n×k, and V is k×n. This can be derived using blockwise matrix inversion.

While the identity is primarily used on matrices, it holds in a general ring or in an Ab-category.