QR factorization using Householder Reflector

Question

For a 3 by 2 matrix, I want to find the QR factorization using Householder Reflector. For example, The first column of A is $(2, 1, 2)$ and the second column of A is $(1, -1, 1)$. Let $H_1$ be $I-2vv^T/(v^Tv)$, I found that the three rows of $H_1A$ are $(3,1), (0,-1),(0,1)$, which suggests that x bar is equal to $(-1,1)$. Now in order to find $H_2$, I need v bar, but to find v bar, I need w bar. Can someone help what is the w bar here? $(v=w-x)$

Answer 1

You are in the right direction except that you have not defined your notation and hence difficult to understand. $$ A = \left[ \begin{array}{rr} 2 & 1 \\ 1 & -1 \\ 2 & 1 \end{array} \right]. $$After the first step, we get $$ H_1 A = \left[ \begin{array}{rr} 3 & 1 \\ 0 & -1 \\ 0& 1 \end{array} \right]. $$ Now, ignore the first row and the column of the matrix $H_1A$ before we apply a Householder reflector to the remainder of the matrix. The important part of $H_1A$ is at the bottom right. $$ H_1A = \left[ \begin{array}{r|r} x & x \\ \hline 0 & -1 \\ 0 & 1 \end{array} \right]. $$ We apply the Householder reflector $H_2$ to the last two rows of $H_1A$ to get $$ \left[ \begin{array}{r|r} x & x \\ \hline 0 & \sqrt{2} \\ 0 & 0 \end{array} \right]. $$ The above is is the $R$ part of the QR-factorization.

If $A$ is a larger matrix, we ignore the first two rows and the first two columns and repeat the same procedure.