12.3. The Householder transformation

Most discussions of vector operations include addition, subtraction, and multiplication, but vector reflection is also a critical operation in many algorithms. The concept is simple: given an input vector and a vector perpendicular to a surface, the goal is to find the reflection of the input vector across the surface. The procedure for computing this reflection is called the Householder transformation, and this section will examine this transformation in detail. But first, it’s important to be familiar with the theory of vector projection.

12.3.1. Vector projection

The dot product of two vectors provides an idea of their relative directions. If the product is positive and large compared to the vectors’ lengths, it implies that the two vectors are pointing in similar directions. If the dot product is negative, it implies that the two vectors are pointing in different directions. If the dot product is 0, it means the two vectors point at right angles to one another.