Mathematics-Online lexicon: Orthogonal Projection

	[home] [lexicon] [problems] [tests] [courses] [auxiliaries] [notes] [staff]
	Mathematics-Online lexicon:
	Orthogonal Projection

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

The orthogonal projection

$\displaystyle x\mapsto P_U(x) \in U$

onto a subspace

is characterised by the following condition of orthogonality:

$\displaystyle \langle x - P_U(x), u \rangle = 0,\quad \forall u\in U\, .$

$\includegraphics[width=0.6\linewidth]{a_orthogonale_projektion}$

If $u_1,\ldots,u_j$ is a orthogonal basis of , then we have

$\displaystyle P_U(x) = \sum_{k=1}^j \frac{\langle x, u_j \rangle}{\Vert u_k\Vert^2} u_j\,.$

automatically generated 3/28/2008