# QR decomposition with lower triangular matrix using Householder reflection

## Problem

Find householder matrices $$H_1,H_2,\cdots,H_n$$ such that

$$H_n\cdots H_1 A = L$$

where $$A$$ : $$n \times n$$ matrix and $$L$$ : $$n \times n$$ lower triangular matrix.

## Try

By defining $$v_k:= [\cdots, sgn(x_k) |x_k|, \cdots]$$ and $$H_k := I - 2v_kv_k^T/v_k^Tv_k$$, we can make

$$H_n\cdots H_1 A = U$$

where $$U$$ : $$n \times n$$ upper triangular matrix

But I'm currently stuck at how to define $$v_k$$ to make RHS lower triangular.

Start from the last column, you are then in almost the same setting as for the $$QR$$ with upper $$R$$: you start with the column that will be transformed to a column with only zeros except for one coefficient (but it will be $$\lambda e_n$$ instead of $$\lambda e_1$$), and continue as usual.
Alternately, you can do that with the same algorithm as usual ($$QR$$ with upper $$R$$), by first reversing the order of the columns and rows of $$A$$, and doing the same afterwards on $$Q$$ and $$R$$. You can apply this to all Householder reflections, and you can check that a "reversed" reflection $$H_k$$ is still a Householder reflection with reversed $$v_k$$.
Note that if $$f$$ is the function that reverses the order of rows and columns, then $$f(AB)=f(A)f(B)$$. I don't know if it has a specific notation.