Inverting the sum of two singular matrices

Abstract

Square matrices of the form $\tilde{A}=A+eD{f}^{\ast }$ are considered. An explicit expression for the inverse is given, provided $\tilde{A}$ and $D$ are invertible with $rank\left(\tilde{A}\right)=rank\left(A\right)+rank\left(eD{f}^{\ast }\right)$. The inverse is presented in two ways, one that uses singular value decomposition and another that depends directly on the components $A$, $e$, $f$ and $D$. Additionally, a matrix determinant lemma for singular matrices follows from the derivations.