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
Canonical name	MatrixInversionLemma
Date of creation	2013-03-22 15:38:44
Last modified on	2013-03-22 15:38:44
Owner	kronos (12218)
Last modified by	kronos (12218)
Numerical id	6
Author	kronos (12218)
Entry type	Result
Classification	msc 47S99
Synonym	Sherman-Morrison formula
Synonym	Woodbury matrix identity
Synonym	Woodbury formula
Synonym	rank-k correction
Related topic	SchurComplement