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Orthogonal Projection


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The orthogonal projection

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

onto a subspace $ U$ 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 $ U$, then we have

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

Examples:


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  automatically generated 3/28/2008