# 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.

The Woodbury matrix identity allows cheap computation of inverses and solutions to linear equations. However, little is known about the numerical stability of the formula. There are no published results concerning its error bounds. Anecdotal evidence ^{[6]} suggests that it may diverge even for seemingly benign examples (when both the original and modified matrices are well-conditioned).