Mathematics-Online lexicon: Normal Equations

	[home] [lexicon] [problems] [tests] [courses] [auxiliaries] [notes] [staff]
	Mathematics-Online lexicon:
	Normal Equations

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

overview

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 $A = n \leq m$ .

see also:

Keyword: Normal equation
Keyword: System of equations, linear: least squares problem

[Annotations] [Examples]

automatically generated 5/23/2011