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Rank-1 Modification of an Inverse Matrix

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Let $ A$ be an invertible matrix and $ B$ the matrix obtained from $ A$ by replacing the $ j$-th column by a vector $ v$. If

$\displaystyle Au=v \,, \quad u_j \neq 0 \,, $

then $ B$ is invertible and

$\displaystyle B^{-1}=Q A^{-1} \,, $

where

$\displaystyle Q=E+ \frac{1}{u_j} \left( e_j -u \right) e_j^{\operatorname t} $

with $ E$ the unit matrix and $ e_j$ the $ j$-th unit vector.
(Authors: Höllig/Pfeil/Walter)

see also:

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  automatically generated 4/24/2007