Householder Reflection

Let $\mathbf{P}_{\underline{u}}$ denote the projection matrix which orthogonally projects vectors onto ${\underline{u}}$, i.e.,

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

and

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

specifically projects $\underline{x}$ onto $\underline{u}$. Since the projection is orthogonal, we have $\left<\underline{x}-\mathbf{P}_{\underline{u}}\underline{x},\underline{x}\right>=\underline{0}$.

• We may interpret $(I-\mathbf{P}_{\underline{u}})\underline{x}$ as the difference vector between $\underline{x}$ and $\mathbf{P}_{\underline{u}}\underline{x}$, its orthogonal projection onto $\underline{u}$, since
  $$(I-\mathbf{P}_{\underline{u}})\underline{x}+ \mathbf{P}_{\underline{u}}\underline{x}= \underline{x}$$
  and we have $(I-\mathbf{P}_{\underline{u}})\underline{x}\perp \underline{x}$ 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 $\underline{u}$:
  $$\underline{y}= \mathbf{P}_{\underline{u}}\underline{x}- (I-\mathbf{P}_{\underline{u}})\underline{x}= (2\mathbf{P}_{\underline{u}}- I)\underline{x}$$

• $\underline{y}$ is obtained by reflecting $\underline{x}$ about $\underline{u}$
• This is called a Householder reflection