These frequently used formulae allow to quickly calculate the inverse of a slight modification of an operator (matrix) $x$ , given that $x^{-1}$ is already known.

The matrix inversion lemma states that $$\left( x+ s \sigma z^* \right)^{-1} = x^{-1} - x^{-1} s \left( \sigma^{-1} + z^* x^{-1} s \right)^{-1} z^* x^{-1},$$ where $x$ , $s$ , $z^*$ and $\sigma$ are operators (matrices) of appropriate size. The formula especially is convenient if the rank of the regular $\sigma$ is 1, or small in comparison to $x$ 's rank.

This identity, involving the inverse of Schur's complement $d- z^* x^{-1} s$ (hopefully this may be easily computed) holds as well: $$\begin{bmatrix} x & s \\ z^* & d \end{bmatrix}^{-1} = \begin{bmatrix} x^{-1}+ x^{-1} s (d- z^* x^{-1} s)^{-1} z^* x^{-1} & -x^{-1} s (d-z^*x^{-1}s)^{-1} \\ -(d-z^*x^{-1}s)^{-1} z^* x^{-1} & (d-z^*x^{-1}s)^{-1} \end{bmatrix}.$$