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Math Help - Householder reflector

  1. #1
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    Householder reflector

    A Householder reflector is a matrix H \in \mathbb{R}^{n \times n} of the form H = I - \alpha \boldsymbol{a}\boldsymbol{a}^T where \boldsymbol{0} \ne \boldsymbol{a} \in \mathbb{R}^n and \displaystyle \alpha = \frac{2}{||\boldsymbol{a}||^2}.

    Given a nonzero vector \boldsymbol{v} \in \mathbb{R}^n, explain how to find a scalar \beta \in \mathbb{R} such that if \boldsymbol{a} = \boldsymbol{v} + \beta \boldsymbol{e}_1 then \displaystyle H \boldsymbol{v} = \begin{bmatrix} -\beta \\ 0 \\ 0 \\ \vdots \\ 0 \end{bmatrix}.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Choose \beta=\left\|{v}\right\| .
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