Editing Woodbury matrix identity (section)

==== Binomial inverse theorem ====

If ''A'', ''B'', ''U'', ''V'' are matrices of sizes ''n''×''n'', ''k''×''k'', ''n''×''k'', ''k''×''n'', respectively, then
<math display="block">
  \left(A + UBV\right)^{-1} =
  A^{-1} - A^{-1}UB\left(B+BVA^{-1}UB\right)^{-1} BVA^{-1}
</math>

provided ''A'' and ''B'' + ''BVA''<sup>−1</sup>''UB'' are nonsingular. Nonsingularity of the latter requires that ''B''<sup>−1</sup> exist since it equals {{nowrap|''B''(''I'' + ''VA''<sup>−1</sup>''UB'')}} and the rank of the latter cannot exceed the rank of ''B''.<ref name=HS>{{cite journal | last1 = Henderson | first1 = H. V. | last2 = Searle | first2 = S. R. | year = 1981 | title = On deriving the inverse of a sum of matrices | url = http://ecommons.cornell.edu/bitstream/1813/32749/1/BU-647-M.pdf| journal = SIAM Review | volume = 23 | issue = 1 | pages = 53–60 | doi = 10.1137/1023004 | jstor = 2029838 | hdl = 1813/32749 | hdl-access = free }}</ref>

Since ''B'' is invertible, the two ''B'' terms flanking the parenthetical quantity inverse in the right-hand side can be replaced with {{nowrap|(''B''<sup>−1</sup>)<sup>−1</sup>,}} which results in the original Woodbury identity.

A variation for when ''B'' is singular and possibly even non-square:<ref name=HS/>
<math display="block">(A + UBV)^{-1} = A^{-1} - A^{-1}U(I + BVA^{-1}U)^{-1}BVA^{-1}.</math>

Formulas also exist for certain cases in which ''A'' is singular.<ref>Kurt S. Riedel, "A Sherman–Morrison–Woodbury Identity for Rank Augmenting Matrices with Application to Centering", ''SIAM Journal on Matrix Analysis and Applications'', 13 (1992)659-662, {{doi|10.1137/0613040}} [http://math.nyu.edu/mfdd/riedel/ranksiam.ps preprint] {{MR|1152773}}</ref>