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

Abstract

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

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