Factorization Algorithm
How does this help us do the factorization? The key observation is that given a vector \(\mathbf{z}\) we can choose a \(\mathbf{V}\) so that \(\mathbf{P}\) reflect \(\mathbf{z}\) onto the \(\mathbf{e}_1\) axis:
\[ \mathbf{P}\mathbf{z} = \begin{bmatrix} \pm \| \mathbf{z} \|\\0 \\ \vdots \\ 0 \end{bmatrix} = \pm \| \mathbf{z} \| \mathbf{e}_1. \]
This uses the fact that \(\mathbf{P}\) is orthogonal and so preserves the norm.
The vector that will do this is:
\[ \mathbf{v} = \frac{\mathbf{w}}{||\mathbf{w}||}\text{, }\mathbf{w} = ||\mathbf{z}||e_1-z \]
The book describes the process in detail, but the essence of the idea is to use this idea to successively turn the matrix \(\mathbf{A}\) into \(\mathbf{R}\). The orthogonal projection matrices form \(\mathbf{Q}\)