matrix inversion lemma

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: