The previous note built from Gram-Schmidt — a natural, direct construction, but (as the closing note of this topic shows) one with a serious numerical flaw. This note builds a second, completely different way — via Householder reflectors, orthogonal matrices that each zero out an entire column below the diagonal in one stroke — and (briefly) via Givens rotations, which zero out one entry at a time. Both constructions reach the same factorisation as Gram-Schmidt; the point of learning a second route is that this one does not suffer Gram-Schmidt's instability.
Learning objectives. By the end of this note you should be able to:
- construct a Householder reflector that sends a given vector to a multiple of ;
- verify a Householder reflector is orthogonal and symmetric;
- explain, at a high level, how a sequence of reflectors builds a full factorisation;
- state what a Givens rotation does differently, and when it is preferred.
In this lesson
- Householder reflectors
- Givens rotations
Before you start
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