Householder Reflections

For completeness, this section derives the Householder reflection matrix from geometric considerations [454]. Let $\mathbf{P}_{\underline{u}}$ denote the projection matrix which orthogonally projects vectors onto ${\underline{u}}$ , i.e.,

$$\mathbf{P}_{\underline{u}}= \frac{\underline{u}\,\underline{u}^T}{\underline{u}^T\underline{u}} = \frac{\underline{u}\,\underline{u}^T}{\left\Vert\,\underline{u}\,\right\Vert^2}$$

and

$$\mathbf{P}_{\underline{u}}\, \underline{x}= \underline{u}\,\frac{\left<\underline{u},\underline{x}\right>}{\left\Vert\,\underline{u}\,\right\Vert^2}$$

specifically projects $\underline {x}$ onto $\underline{u}$ . Since the projection is orthogonal, we have

$$\left<\underline{x}-\mathbf{P}_{\underline{u}}\underline{x},\underline{u}\right>=\left<(\mathbf{I}-\mathbf{P}_{\underline{u}})\underline{x},\underline{u}\right>=\underline{0}.$$

We may interpret $(\mathbf{I}-\mathbf{P}_{\underline{u}})\underline{x}$ as the difference vector between $\underline {x}$ and $\mathbf{P}_{\underline{u}}\underline{x}$ , its orthogonal projection onto $\underline{u}$ , since

$$(\mathbf{I}-\mathbf{P}_{\underline{u}})\underline{x}+ \mathbf{P}_{\underline{u}}\underline{x}= \underline{x}$$

and we have $(\mathbf{I}-\mathbf{P}_{\underline{u}})\underline{x}\perp \mathbf{P}_{\underline{u}}\underline{x}$ by definition of the orthogonal projection. Consequently, the projection onto $\underline{u}$ minus this difference vector gives a reflection of the vector $\underline {x}$ about $\underline{u}$ :

$$\underline{y}= \mathbf{P}_{\underline{u}}\underline{x}- (\mathbf{I}-\mathbf{P}_{\underline{u}})\underline{x}= (2\mathbf{P}_{\underline{u}}- \mathbf{I})\underline{x}$$

Thus, $\underline{y}$ is obtained by reflecting $\underline {x}$ about $\underline{u}$ --a so-called Householder reflection.