# Woodbury matrix identity

## Theorem of matrix ranks / From Wikipedia, the free encyclopedia

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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]

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.