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][1] 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.


  [1]: http://en.wikipedia.org/wiki/Sherman%E2%80%93Morrison_formula