Inverting the sum of two singular matrices
Sofia Eriksson, Jonas Nordqvist · Results in Applied Mathematics · 2024
Square matrices of the form A ˜ = A + e D f ∗ are considered. An explicit expression for the inverse is given, provided A ˜ and D are invertible with rank ( A ˜ ) = rank ( A ) + rank ( e D f ∗ ) . 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.