Householder reflector

**math2011** · March 27th 2011, 01:50 AM

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 $\displaystyle H \boldsymbol{v} = \begin{bmatrix} -\beta \\ 0 \\ 0 \\ \vdots \\ 0 \end{bmatrix}$ .

**FernandoRevilla** · March 27th 2011, 02:23 AM

Choose $\beta=\left\|{v}\right\|$ .

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