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


where x, s, z* and σ are operators (matrices) of appropriate size. The formula especially is convenient if the rank of the regular σ is 1, or small in comparison to x’s rank.

This identity, involving the inverse of Schur’s complement d-z*x-1s (hopefully this may be easily computed) holds as well:

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