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

Matrices of the form bfA + (bfV1 + bfW1)bfG(bfV2 + bfW2)^* are considered where bfA is a singular ℓ × ℓ matrix and bfG is a nonsingular k × k matrix, k ≤ ℓ. Let the columns of bfV1 be in the column space of bfA and the columns of bfW1 be orthogonal to bfA. Similarly, let the columns of bfV2 be in the column space of bfA^* and the columns of bfW2 be orthogonal to bfA^*. An explicit expression for the inverse is given, provided that bfWi^* bfWi has rank k. %and bfW1 and bfW2 have the same column space. An application to centering covariance matrices about the mean is given.