Editing Woodbury matrix identity (section)

== Discussion ==
To prove this result, we will start by proving a simpler one.  Replacing ''A'' and ''C'' with the [[identity matrix]] ''I'', we obtain another identity which  is a bit simpler:
<math display="block"> \left(I + UV \right)^{-1} = I - U \left(I + VU \right)^{-1} V. </math>
To recover the original equation from this ''reduced identity'', replace <math>U</math> by <math>A^{-1}U</math> and <math>V</math> by <math>CV</math>.

This identity itself can be viewed as the combination of two simpler identities. We obtain the first identity from
<math display="block"> I = (I + P)^{-1}(I + P) = (I + P)^{-1} + (I + P)^{-1}P,</math>
thus,
<math display="block"> (I + P)^{-1} = I-(I + P)^{-1}P,</math>
and similarly
<math display="block"> (I + P)^{-1} = I - P (I + P)^{-1}.</math>
The second identity is the so-called '''push-through identity'''<ref name="HS"/>
<math display="block"> (I + UV)^{-1} U = U (I + VU)^{-1} </math>
that we obtain from
<math display="block"> U(I + VU)=(I + UV)U</math>
after multiplying by <math>(I + VU)^{-1}</math> on the right and by <math>(I + UV)^{-1}</math> on the left.

Putting all together,
<math display="block"> \left(I + UV \right)^{-1} = I - UV \left(I + UV \right)^{-1} = I - U \left(I + VU \right)^{-1} V. </math>
where the first and second equality come from the first and second identity, respectively.
=== Special cases ===

When <math>V, U</math> are vectors, the identity reduces to the [[Sherman–Morrison formula]].

In the scalar case, the reduced version is simply
<math display="block">\frac{1}{1 + uv} = 1 - \frac{uv}{1 + vu}.</math>

==== Inverse of a sum ====

If ''n'' = ''k'' and ''U'' = ''V'' = ''I''<sub>''n''</sub> is the identity matrix, then

<math display="block">\begin{align}
\left(A + B\right)^{-1} &= A^{-1} - A^{-1} \left(B^{-1} + A^{-1}\right)^{-1} A^{-1} \\[1ex]
&= A^{-1} - A^{-1} \left(A B^{-1} + {I}\right)^{-1}.
\end{align}</math>

Continuing with the merging of the terms of the far right-hand side of the above equation results in [[Hua's identity]]
<math display="block">\left({A} + {B}\right)^{-1} = {A}^{-1} - \left({A} + {A}{B}^{-1}{A}\right)^{-1}.</math>

Another useful form of the same identity is
<math display="block">\left({A} - {B}\right)^{-1} = {A}^{-1} + {A}^{-1}{B}\left({A} - {B}\right)^{-1},</math>

which, unlike those above, is valid even if <math>B</math> is [[singular matrix|singular]], and has a recursive structure that yields
<math display="block">\left({A} - {B}\right)^{-1} = \sum_{k=0}^{\infty} \left({A}^{-1}{B}\right)^k{A}^{-1}</math>
if the [[spectral radius]] of <math>A^{-1}B</math> is less than one.  That is, if the above sum converges then it is equal to <math>(A-B)^{-1}</math>.

This form can be used in perturbative expansions where ''B'' is a perturbation of ''A''.

=== Variations ===

==== 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>

==== Pseudoinverse with positive semidefinite matrices ====

In general Woodbury's identity is not valid if one or more inverses are replaced by [[Moore–Penrose inverse|(Moore–Penrose) pseudoinverses]]. However, if <math>A</math> and <math>C</math> are [[Positive semidefinite matrices|positive semidefinite]], and <math>V = U^\mathrm H</math> (implying that <math>A + UCV</math> is itself positive semidefinite), then the following formula provides a generalization:<ref>{{cite book |last1=Bernstein |first1=Dennis S. |title=Scalar, Vector, and Matrix Mathematics: Theory, Facts, and Formulas |date=2018 |publisher=Princeton University Press |location=Princeton |isbn=9780691151205 |page=638 |edition=Revised and expanded}}</ref><ref>{{cite book |last1=Schott |first1=James R. |title=Matrix analysis for statistics |date=2017 |publisher=John Wiley & Sons, Inc. |location=Hoboken, New Jersey |isbn=9781119092483 |page=219 |edition=Third}}</ref>
<math display="block">
\begin{align}
\left(XX^\mathrm H + YY^\mathrm H\right)^+
&= \left(ZZ^\mathrm H\right)^+ + \left(I - YZ^+\right)^\mathrm H X^{+\mathrm H} E X^+ \left(I - YZ^+\right), \\
Z &= \left(I - XX^+\right) Y, \\
E &= I - X^+Y \left(I - Z^+Z\right) F^{-1} \left(X^+Y\right)^\mathrm H, \\
F &= I + \left(I - Z^+Z\right) Y^\mathrm H \left(XX^\mathrm H\right)^+ Y \left(I - Z^+Z\right),
\end{align}
</math>

where <math>A + UCU^\mathrm H</math> can be written as <math>XX^\mathrm H + YY^\mathrm H</math> because any positive semidefinite matrix is equal to <math>MM^\mathrm H</math> for some <math>M</math>.