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, $k \le \ell$. 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 $k$. %and $\bf{W}_1$ and $\bf{W}_2$ have the same column space. An application to centering covariance matrices about the mean is given.