TITLE: Sherman-Morrison type formula for Moore-Penrose pseudoinverse
QUESTION [14 upvotes]: Given an $n\times n$ invertible matrix $\mathbf A$ and two column vectors $\mathbf u$, $\mathbf v\in\mathbb R^n$, suppose that $1 + {\mathbf v}^T {\mathbf A}^{-1}\mathbf u \neq 0$.
Then the Sherman-Morrison formula states that
\begin{equation*}
(\mathbf A + \mathbf u \mathbf v^T)^{-1} =
\mathbf A^{-1} -
{\mathbf A^{-1}\mathbf u\mathbf v^T \mathbf A^{-1} \over 1 + \mathbf v^T \mathbf A^{-1}\mathbf u}.
\end{equation*}
Question: I'm wondering whether we have a similar formula when the inverse in the Sherman-Morrison formula is replaced by the Moore-Penrose pseudoinverse in case that $\mathbf A$ is singular matrix.

REPLY [12 votes]: It's all in Meyer's paper Generalized Inversion of Modified Matrices published 1973 in SIAM Journal on Applied Mathematics.
The material is also available at around p.51 in the Meyer & Campbell book.