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.
Best Answer
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.