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 \(\mathbf{v}\)