# Householder reflector

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• Mar 27th 2011, 01:50 AM
math2011
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 $\displaystyle H \boldsymbol{v} = \begin{bmatrix} -\beta \\ 0 \\ 0 \\ \vdots \\ 0 \end{bmatrix}$.
• Mar 27th 2011, 02:23 AM
FernandoRevilla
Choose $\beta=\left\|{v}\right\|$ .