Square matrices of the form $\widetilde{ \mathbf{ A}} =\mathbf{ A} + \mathbf{ e}D \mathbf{ f}^*$ are considered. An explicit expression for the inverse is given, provided $\widetilde{ \mathbf{ A}}$ and $D$ are invertible with $\text{ rank}(\widetilde{ \mathbf{ A}}) =\text{ rank}(\mathbf{ A})+\text{ rank}(\mathbf{ e}D \mathbf{ f}^*)$. The inverse is presented in two ways, one that uses singular value decomposition and another that depends directly on the components $\mathbf{ A}$, $\mathbf{ e}$, $\mathbf{ f}$ and $D$. Additionally, a matrix determinant lemma for singular matrices follows from the derivations.
Categories: math.NA cs.NA math.RA