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

Matrices of the form ${\bf A} + ({\bf V}_1 + {\bf W}_1 ) {\bf G} ({\bf V}_2 + {\bf W}_2 )^ * $ are considered where ${\bf A}$ is a singular$\ell \times \ell $ matrix and ${\bf G}$ is a nonsingular $k \times k$ matrix, $k \leq \ell $. Let the columns of ${\bf V}_1 $ be in the column space of ${\bf A}$ and the columns of ${\bf W}_1 $ be orthogonal to ${\bf A}$. Similarly, let the columns of ${\bf V}_2 $ be in the column space of ${\bf A}^ * $ and the columns of ${\bf W}_2 $ be orthogonal to ${\bf A}^ * $. An explicit expression for the inverse is given, provided that ${\bf W}_i^ * {\bf W}_i $ has rank k. An application to centering covariance matrices about the mean is given.