QR factorisation

factorisation of a matrix is
where is a square orthogonal matrix
and is upper triangular.
If , we can have a reduced factorisation
with being and being ,
where the columns of are orthogonal.

One method for obtaining the factorisation
is the classic Gram-Schmidt process, where

with and for and .

Givens rotation

Use Givens Rotation to introduce s from left to right.

Householder reflections

Take a Householder Reflection
where for
and
We find where .
Then:

Thus doesn’t affect the first rows and columns of a matrix,
and if is the -th column, we have so we introduced s to the -th column.