Householder Transformation

In linear algebra, a Householder transformation (also known as a Householder reflection or elementary reflector) is a linear transformation that describes a reflection about a plane or hyperplane containing the origin.

H = I - 2 w w^{T}

Where $w$ is a unit vector and $I$ is the identity matrix.

How does it work?

Let's take the 3-dimensional space as an example. As shown in the figure above, we think of $w$ as a normal vector of a plane. Suppose we have a given vector $v$ , we decompose $v$ onto a set of orthogonal basis $\{v_x, v_y, v_z\}$ , ensuring that $v_x = w$ . This decomposition meets:

\begin{aligned} w\cdot v_x &= 0\\ w\cdot v_y &= 0\end{aligned}

Then, after the Householder transformation, $v$ will become:

\begin{aligned} Hv &= (I - 2 w w^T) (v_x + v_y + v_z)\\ &= v_x - 2 w w^T v_x + v_y - 2 w w^T v_y + v_z - 2 w w^T v_z\\ &= -v_x + v_y + v_z\\ &= v' \\ \end{aligned}

Thus, $v$ is mirrored onto $v'$ along the normal vector of $w$ .