Mathematics-Online lexicon: Normal Equations

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	Mathematics-Online lexicon:
	Normal Equations

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For an $m \times n$ matrix , any solution of the least squares problem $\Vert Ax-b\Vert _2$ satisfies the normal equations

$\displaystyle A^{\operatorname t}Ax = A^{\operatorname t}b.$

$\includegraphics[width=.4\moimagesize]{a_normalengleichungen}$

Geometrically, this means that the residuum is orthogonal to the columns of , i.e. to the subspace $\operatorname{im} A$ of $\mathbb{R}^m$ . The solution is unique if rank $A = n \leq m$ .

Annotation:

Proof: Normal Equations

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automatically generated 5/23/2011