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.