Inverting the sum of two singular matrices

Abstract

Square matrices of the form $\tilde{\mathbf{A}}=\mathbf{A}+\mathbf{e}D{\mathbf{f}}^{\ast }$ are considered. An explicit expression for the inverse is given, provided $\tilde{\mathbf{A}}$ and $D$ are invertible with $\text{rank}(\tilde{\mathbf{A}})=\text{rank}(\mathbf{A})+\text{rank}(\mathbf{e}D{\mathbf{f}}^{\ast })$. 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.