The Woodbury matrix identity

The Woodbury matrix identity is a useful identity in linear algebra. It says that you can invert the sum of a matrix plus a \(k\)-rank correction by doing a rank \(k\)-correction to the inverse of the original matrix. It is also called matrix inversion lemma or Sherman-Morrison-Woodbury formula.

More explicitly the identity states:

\[\left( \mathbf{A} + \mathbf{U}\mathbf{C}\mathbf{V} \right)^{-1} = \mathbf{A}^{-1} - \mathbf{A}^{-1} \mathbf{U}\left( \mathbf{C}^{-1} + \mathbf{V} \mathbf{A}^{-1} \mathbf{U} \right)^{-1} V \mathbf{A}^{-1}\]

where \(\mathbf{A},\mathbf{U},\mathbf{C},\mathbf{V}\) are all matrices with the correct shapes, specifically, \(\mathbf{A}\) is a square \(n\times n\), \(\mathbf{U}\) is \(n \times k\), \(\mathbf{C}\) is \(k\times k\) and \(\mathbf{V}\) is \(k \times n\).

A more general form of the Woodbury matrix identity can be found using blockwise matrix inversion.