Inverting the sum of two singular matrices
<|reference_start|>Inverting the sum of two singular matrices: 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.<|reference_end|>
@article{eriksson2024inverting,
    title={Inverting the sum of two singular matrices},
    author={Sofia Eriksson and Jonas Nordqvist},
    journal={arXiv preprint arXiv:2403.16896},
    year={2024},
    archivePrefix={arXiv},
    eprint={2403.16896},
    primaryClass={math.NA cs.NA math.RA}
}