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.