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).
^Max A. Woodbury, Inverting modified matrices, Memorandum Rept. 42, Statistical Research Group, Princeton University, Princeton, NJ, 1950, 4pp MR38136
^Max A. Woodbury, The Stability of Out-Input Matrices. Chicago, Ill., 1949. 5 pp. MR32564
^Guttmann, Louis (1946). "Enlargement methods for computing the inverse matrix". Ann. Math. Statist. 17 (3): 336–343. doi:10.1214/aoms/1177730946.
^Hager, William W. (1989). "Updating the inverse of a matrix". SIAM Review. 31 (2): 221–239. doi:10.1137/1031049. JSTOR 2030425. MR 0997457.
^Higham, Nicholas (2002). Accuracy and Stability of Numerical Algorithms (2nd ed.). SIAM. p. 258. ISBN 978-0-89871-521-7. MR 1927606.
^
"MathOverflow discussion". MathOverflow.
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