the pseudoinverse of a matrix with multiple rank one update

Question

Generalized Inverses and Ranks of Block Matrices by Carl Meyer provides the formula for $(A+cd^T)\dagger$, where $c,d^T$ are column vectors. Is there a general formula for $(A+c_1d_1^T+c_2d_2^T)\dagger$?

Any leads would be appreciated!

Answer 1

Note that $$u_1v_1^T+u_2v_2^T= UV^T,$$ where $U = [u_1, u_2]$ and $V=[v_1, v_2]$ (i.e., $U$ has $u_1$ and $u_2$ as columns, and same for $V$).

Then if everything that need to be invertible is invertible, the Woodbury identity gives you what you want:

$$(A+UV^T)^{-1} = A^{-1}-A^{-1}U(I+V^TA^{-1}U)^{-1}V^TA^{-1}.$$

However, I'm not sure about pseudoinverses.