Matrix inversion lemma
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 (conformable) sizes. 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.
For a more general formula for which the matrix C need not be invertible or even square, see Binomial inverse theorem.
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. For mere numbers, the identity reduces to the following relation of fractions
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