A Sherman–Morrison–Woodbury Identity for Rank Augmenting Matrices with Application to Centering

Matrices of the form A + ( V₁ + W₁ ) G ( V₂ + W₂ )^ * are considered where A is a singular× matrix and G is a nonsingular k × k matrix, k ≤. Let the columns of V₁ be in the column space of A and the columns of W₁ be orthogonal to A. Similarly, let the columns of V₂ be in the column space of A^ * and the columns of W₂ be orthogonal to A^ *. An explicit expression for the inverse is given, provided that Wᵢ^ * Wᵢ has rank k. An application to centering covariance matrices about the mean is given.