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:

 $\begin{bmatrix}x&s\\ z^{*}&d\end{bmatrix}^{-1}=\begin{bmatrix}x^{-1}+x^{-1}s(d-z^{*}x^{-1}s)^{-1}z^% {*}x^{-1}&-x^{-1}s(d-z^{*}x^{-1}s)^{-1}\\ -(d-z^{*}x^{-1}s)^{-1}z^{*}x^{-1}&(d-z^{*}x^{-1}s)^{-1}\end{bmatrix}.$
Title matrix inversion lemma MatrixInversionLemma 2013-03-22 15:38:44 2013-03-22 15:38:44 kronos (12218) kronos (12218) 6 kronos (12218) Result msc 47S99 Sherman-Morrison formula Woodbury matrix identity Woodbury formula rank-k correction SchurComplement