3.4 Computing QR factorizations

QR factorization can be computed with Gram-Schmidt process. This section of the book shows how this is down mechanically using Householder reflections.

Householder reflections

• Householder reflector is a matrix of the form: $\mathbf{P} = \mathbf{I} - 2 \mathbf{v}\mathbf{v}^T$ where $v$ is a unit vector.

• Note that $\mathbf{P}$ is orthogonal and for any vector $\mathbf{x}$:

$$\mathbf{P}\mathbf{x} = \mathbf{x} - 2 \mathbf{v} (\mathbf{v}^T\mathbf{x})$$

This is a reflection of $\mathbf{x}$ about the hyperplane with normal vector $\mathbf{v}$