Mathematics-Online lexicon: Solution of a Symmetric Positive Linear System

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	Mathematics-Online lexicon:
	Solution of a Symmetric Positive Linear System

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

A linear system

with symmetric positive definite matrix

can be solved with the aid of the Cholesky factorization $S = R^{\operatorname t} R$ :

$\displaystyle R^{\operatorname t} \underbrace{Rx}_y =b \; \Longleftrightarrow \; R^{\operatorname t} y=b \; \wedge \; Rx=y \,.$

The solutions

and

of the two triangular systems are determined via forward and backward substitution, respectively.

In particular, the Cholesky factorization can be used for solving the normal equations

$\displaystyle \underbrace{A^{\operatorname t}A}_S x = \underbrace{A^{\operatorname t}c}_b$

if the $m\times n$ matrix

has maximal rank

(Authors: Höllig/Pfeil/Walter)

see also:

automatically generated 4/24/2007