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}

Q-less QR and least squares

  • Since we only need \mathbf{Q} to compute \mathbf{Q}^T b, we don’t need the full \mathbf{Q}

  • Leads to “Q-less” factorization: In julia a special “QRCompactWYQ” object is returned for efficient calculation of \mathbf{Q}^T b