# Sherman-Morrison type formula for Moore-Penrose Pseudoinverse

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.

• Have a look at Mikael's answer at: mathoverflow.net/questions/72059/… – Suvrit Nov 3 '13 at 18:38
• If $\mathbf A$ is symmetric and so is the update to it, then I get that the Sherman-Morrison formula works as is (replacing inverse with pseudo-inverse of course). Otherwise, if I am correct, the formula gives you only a general inverse, and correction using the null space is required to make it the desired pseudo-inverse. – adam W Nov 4 '13 at 6:38

It's all in Meyer's 1973 paper:

http://www.jstor.org/discover/10.2307/2099767?uid=3738240&uid=2&uid=4&sid=21103607586533

The material is also available at around p.51 in the Meyer & Campbell book.

Theorem 15 in the following paper may be useful for you.

Sujit Kumar Mitra and P. Bhimasankaram, Generalized Inverses of Partitioned Matrices and Recalculation of Least Squares Estimates for Data or Model Changes, Sankhyā: The Indian Journal of Statistics,
Vol. 33, No. 4, Dec., 1971 , page 395-410.

This might be useful: http://en.wikipedia.org/wiki/Moore-Penrose_pseudoinverse#Updating_the_pseudoinverse

"Similarly, it is possible to update the Cholesky factor when a row or column is added, without creating the inverse of the correlation matrix explicitly. However, updating the pseudoinverse in the general rank-deficient case is much more complicated."