Editing Woodbury matrix identity (section)

{{Use American English|date = January 2019}}
{{Short description|Theorem of matrix ranks}}
In [[mathematics]], specifically [[linear algebra]], the '''Woodbury matrix identity''' – named after [[Max A. Woodbury]]<ref>Max A. Woodbury, ''Inverting modified matrices'', Memorandum Rept. 42, Statistical Research Group, Princeton University, Princeton, NJ, 1950, 4pp {{MR|38136}}</ref><ref>Max A. Woodbury, ''The Stability of Out-Input Matrices''. Chicago, Ill., 1949. 5 pp. {{MR|32564}}</ref> – says that the inverse of a rank-''k'' correction of some [[matrix (mathematics)|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.<ref name="guttman">{{cite journal
 |first=Louis |last=Guttmann
 |title=Enlargement methods for computing the inverse matrix
 |journal=Ann. Math. Statist.
 |volume=17 |year=1946 |pages=336&ndash;343 |issue=3
 |doi=10.1214/aoms/1177730946
|doi-access=free}}</ref><ref name="hager">{{cite journal
 |first=William W. |last=Hager
 |title=Updating the inverse of a matrix
 |journal=SIAM Review
 |volume=31 |year=1989 |pages=221&ndash;239 |issue=2
 |doi=10.1137/1031049 |mr=997457 | jstor = 2030425
}}</ref>

The Woodbury matrix identity is<ref name="higham">{{Cite book | last1=Higham | first1=Nicholas | author1-link=Nicholas Higham | title=Accuracy and Stability of Numerical Algorithms | url=https://archive.org/details/accuracystabilit00high_878 | url-access=limited | publisher=[[Society for Industrial and Applied Mathematics|SIAM]] | edition=2nd | isbn=978-0-89871-521-7 | year=2002 | page=[https://archive.org/details/accuracystabilit00high_878/page/n288 258] |mr=1927606 }}
</ref>
<math display="block"> \left(A + UCV \right)^{-1} = A^{-1} - A^{-1}U \left(C^{-1} + VA^{-1}U \right)^{-1} VA^{-1}, </math>

where ''A'', ''U'', ''C'' and ''V'' are [[conformable matrix|conformable matrices]]: ''A'' is ''n''×''n'', ''C'' is ''k''×''k'', ''U'' is ''n''×''k'',  and ''V'' is ''k''×''n''. This can be derived using [[invertible matrix#Blockwise inversion|blockwise matrix inversion]].

While the identity is primarily used on matrices, it holds in a general [[ring (mathematics)|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<ref>
{{cite web 
| url = https://mathoverflow.net/questions/80340/special-considerations-when-using-the-woodbury-matrix-identity-numerically
| title = MathOverflow discussion
| website = MathOverflow
}}
</ref> suggests that it may diverge even for seemingly benign examples (when both the original and modified matrices are [[well-conditioned]]).