Inverting the sum of two singular matrices

Abstract: 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.