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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.

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    $\begingroup$ Have a look at Mikael's answer at: mathoverflow.net/questions/72059/… $\endgroup$
    – Suvrit
    Commented Nov 3, 2013 at 18:38
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    $\begingroup$ 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. $\endgroup$
    – adam W
    Commented Nov 4, 2013 at 6:38
  • $\begingroup$ @adamW Can you provide a reference or a line of reasonment for the fact that the Sherman-Morrison formula applies to the pseudoinverse in case of both $A$ and the update being symmetric? I'm in the need of such a formula, but I have not found some clear reference to such a statement yet. $\endgroup$
    – trenta3
    Commented Nov 24, 2021 at 10:35
  • $\begingroup$ @trenta3 I remember I might have some perinent posts on my math.stackexchange. I haven't posted too terribly much so it may be of use to you to look. Otherwise, if I think of what I was talking about here, I may be able to explain better later. no source-- I think I was doing it on my own really $\endgroup$
    – adam W
    Commented Nov 24, 2021 at 16:30
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    $\begingroup$ @trenta3 one post of mine that talks about psuedo inverse and span: math.stackexchange.com/a/317053/43193, just look for the section proof $\endgroup$
    – adam W
    Commented Nov 26, 2021 at 19:01

4 Answers 4

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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.

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    $\begingroup$ the name of the article is "Generalized Inversion of Modified Matrices" and a stable link is jstor.org/stable/2099767 $\endgroup$
    – LudvigH
    Commented Jan 12, 2021 at 10:00
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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.

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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."

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Schott (2017), "Matrix Analysis for Statistics", 3rd ed., Section 5.6, gives a Woodbury-like expression for $(UU' + VV')^+$.

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