Editing Woodbury matrix identity (section)

==== 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>.