In mathematics (specifically linear algebra), the Woodbury matrix identity, named after Max A. Woodbury 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.
The Woodbury matrix identity is
where A, U, C and V all denote matrices of the correct size. Specifically, A is n-by-n, U is n-by-k, C is k-by-k and V is k-by-n. This can be derived using blockwise matrix inversion.
In the special case where C is the 1-by-1 unit matrix, this identity reduces to the Sherman–Morrison formula. In the special case when C is the identity matrix I, the matrix is known in numerical linear algebra and numerical partial differential equations as the capacitance matrix.
Read more about Woodbury Matrix Identity: Derivation Via Blockwise Elimination, Derivation From LDU Decomposition, Direct Proof, Applications
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