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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\,. $

see also:

[Examples]
  automatically generated 3/28/2008