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Normal Equations

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For an $ m \times n$ matrix $ A$ , any solution $ x$ 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 $ Ax-b$ is orthogonal to the columns of $ A$ , i.e. to the subspace $ \operatorname{im} A$ of $ \mathbb{R}^m$ . The solution is unique if rank $ A = n \leq m$ .

Examples:

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  automatically generated 5/23/2011