I have a QR factorisation question, here with the solution but i do not understand exactly what has been done. Could someone please explain?
$A=\begin{pmatrix}1 & 0 & 3 \\2 & -6 & 3 \\-2 & 3 & -3\end{pmatrix}$
Solution says:
The first householder matrix $H(w_1)=I-2w_1{w^T}_1$ should map the first column vector $(1,2,-2)^T$ onto $||(1,2,-2)^T||_2e_1=3e_1$
Why is it $3e_1$?
It then says, thus we choose
$w_1=\frac{v_1}{||v||_1}$ with $ v_1=\begin{bmatrix}1\\2\\-2\end{bmatrix}-3{\begin{bmatrix}1\\2\\-2\end{bmatrix}}=\begin{bmatrix}-2\\2\\-2\end{bmatrix}$
and thus $w_1= \frac{1}{2 \sqrt 3}\begin{bmatrix}-2\\2\\-2\end{bmatrix}=\frac{1}{ \sqrt 3}\begin{bmatrix}-1\\1\\-1\end{bmatrix}$
The rest of the solution i can see and understand what is going on but i just dont know how they got there. Could someone please explain where the 3 got here $||(1,2,-2)^T||_2e_1=3e_1$ in simple terms and why there is a square root of 3? Thanks