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Givens Transformation

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A Givens transformation

$\displaystyle x \mapsto \left(\begin{array}{rr} c & -s \\ s & c \end{array}\right) \left(\begin{array}{c} x_1 \\ x_2 \end{array}\right) $

with

$\displaystyle c$ $\displaystyle = \sigma v_1 / \Vert v \Vert _2 \,, \quad s =-\sigma v_2 / \Vert v \Vert _2$    
$\displaystyle \sigma$ $\displaystyle = \left\{ \begin{array}{ll} \text{sign} \left( v_1 \right)\, , & \text{if }v_1 \neq 0 \\ 1 \,, & \text{if } v_1 =0 \end{array} \right.$    

is a rotation which maps $ v=\left( v_1 , v_2 \right)^t$ to $ \left( \sigma \Vert v \Vert _2, 0 \right)^t$.

Typically, a Givens transformation is applied to two rows of a matrix:

$\displaystyle \left( \begin{array}{c} p_1 \ldots p_m \\ q_1 \ldots q_m \end{a... ... \right) \to \left( \begin{array}{c} cp-sq \\ sp+cq \end{array} \right) \,. $

The parameters are based on $ v= \left( p_1, \, q_1 \right)^t$ so that the entry $ q_1$ is annihilated by the transformation.

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  automatically generated 3/ 8/2007