The Householder transformation \(H=I-2\frac{\vec{v}\vec{v}^T}{\vec{v}^T\vec{v}}\) is a norm-conserving transformation that effectively reflects each column vector of a matrix through the plane defined by the normal vector \(\vec{v}\).

An important application of Householder transformation is to introduce zeros into a vector by reflecting it onto a column vector of the identity matrix \(\vec{e}\). To find the plane of reflection for this purpose, we require \(H \vec{a} = \alpha \vec{e}\) and \(|H \vec{a}| = |\vec{a}|\)

$$ H \vec{a} = \alpha \vec{e} \Rightarrow \vec{a}-(2\frac{\vec{v}^T\vec{a}}{\vec{v}^T\vec{v}})\vec{v}=\alpha\vec{e} \Rightarrow \vec{v}=(\vec{a}-\alpha\vec{e})/\beta,$$

since we only need the direction of the normal vector \(\vec{v}\) to define the plane of reflection, we may choose \(\vec{v}=\vec{a}-\alpha\vec{e}\). In addition, the transformation should conserve the norm of the vector \(\vec{a}\). Thus \(\alpha=|\vec{a}|\), the sign of \(\alpha\) is conventionally chosen to avoid cancellation.