Topic 1 closed with Fact 1's observation that Gram-Schmidt is secretly a matrix factorisation. This topic makes that factorisation, , the object of direct study: two genuinely different constructions of it (Gram-Schmidt itself, and the numerically superior Householder reflectors), its main application (least squares, solved via orthogonal projection rather than matrix inversion), and the numerical failure mode that makes the choice between the two constructions consequential in practice, not merely academic.
By the end of this topic you should be able to:
- construct via Gram-Schmidt and via Householder reflectors, and state why they differ numerically;
- set up and solve a least-squares problem as an orthogonal projection;
- derive the normal equations and solve them via instead of forming explicitly;
- diagnose Gram-Schmidt's orthogonality loss and know when to reach for a more stable alternative.
In this lesson
- Contents
- The big picture
The full lesson is for members.
Read the full lessonNew to SelfMadeQuant.com? See membership options or browse the Linear Algebra, Optimisation and Numerical Methods syllabus.