The Woodbury matrix identity is \begin{equation} (A+UCV)^{-1}=A^{-1}-A^{-1}U(C^{-1}+VA^{-1}U)^{-1}VA^{-1}. \end{equation} This formula suppose that $A$, $(A+UCV)$ and $(C^{-1}+VA^{-1}U)$ are invertible. What happens to this formula when $A$ is not invertible please? Do you know a formula for that case please? I have read the paper " A Sherman Morrison Woodbury Identity for Rank Augmenting Matrices with Application to Centering " of Kurt S. Riedel but his formula looks so complicated. Thanks.
2026-03-27 13:47:21.1774619241
What happens to woodbury matrix identity when A is not invertible?
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