Every earlier note in this topic treated Gram-Schmidt and Householder as two mathematically equivalent routes to . This closing note explains why, in practice, they are not equally trustworthy: classical Gram-Schmidt, run in floating-point arithmetic on nearly dependent columns, loses orthogonality catastrophically — the computed can fail to be orthogonal by an amount far larger than machine precision would suggest. This note names the failure mode, gives the partial fix (modified Gram-Schmidt), and introduces rank-revealing for the case where 's columns are not just ill-conditioned but genuinely (numerically) dependent.
Learning objectives. By the end of this note you should be able to:
- describe why classical Gram-Schmidt loses orthogonality under rounding error;
- distinguish classical from modified Gram-Schmidt and explain why the latter is more stable;
- recognise when a matrix's columns are numerically rank-deficient from its or structure;
- state what "rank-revealing" adds to an ordinary factorisation.
In this lesson
- Why classical Gram-Schmidt loses orthogonality
- Rank-revealing
Before you start
The full lesson, with worked examples and exercise solutions, is for members.
Read the full lessonNew to SelfMadeQuant.com? See membership options or browse the Linear Algebra, Optimisation and Numerical Methods syllabus.