Factorization Algorithm
How does this help us do the factorization? The key observation is that given a vector z we can choose a V so that P reflect z onto the e1 axis:
Pz=[±‖
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}