Mathematics-Online lexicon: Rank-1 Modification of an Inverse Matrix

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	Mathematics-Online lexicon:
	Rank-1 Modification of an Inverse Matrix

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

Let

be an invertible matrix and

the matrix obtained from

by replacing the

-th column by a vector

. If

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

then

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

the unit matrix and

the

-th unit vector.

(Authors: Höllig/Pfeil/Walter)

Example:

automatically generated 4/24/2007