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Householder-transformation

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A Householder transformation

$\displaystyle x \mapsto Qx = x - \frac{1}{r} (d^t x) d $

with

\begin{displaymath} \begin{array}{rcl} d &=& (\begin{array}{cccc} c_1+\sigma... ...ight. \\ r &=& \vert d_1\vert\,\Vert c\Vert _2 \end{array} \end{displaymath}

is a reflection which maps the vector $ c$ to $ -\sigma\Vert c\Vert _2 e_1 $, i.e. a multiple of the first unit vector.

Typically, a Householder transformation is applied simultaneously to the columns of a matrix:

$\displaystyle A \mapsto A -d \left(d^t A \right)/r $

with $ c=A(:,1)$. This annihilates the entries $ a_{2,1} ,a_{3,1}, \ldots $.

Example:

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  automatically generated 3/ 8/2007