Householder transformation to a vector

Question

I need to know how to find the reflector H of Householder such that if $v= \begin{pmatrix} 3\\ 4\\ 12 \end{pmatrix}$ then $Hv=\begin{pmatrix} 0\\ 0\\ 13 \end{pmatrix}$

I've been reading a bit about it, but usually they focus on removing all the components except the first one, but it's not my case. Could someone help me solve this and give me a simple way to understand this?

Thanks!

Edit:

Let me tell you what I have:

my Householder vector u is $\begin{pmatrix} 3 \\ 4 \\ 12\end{pmatrix} - \begin{pmatrix} 0 \\ 0 \\ 13\end{pmatrix} = \begin{pmatrix} 3 \\ 4 \\ -1\end{pmatrix}$ , then, I use it to build $$H=I-2\frac{u \times u^T}{u^Tu}$$ , but at the end, when I multiply $$, I don't get what I want. What am I doing wrong?

Answer 1

Your method is correct. It must be just a careless computational mistake. Here is an octave output.

octave:2> u = [3; 4; -1];
octave:3> H = eye(3)-2*(u*u')/(u'*u)
H =

   0.30769  -0.92308   0.23077
  -0.92308  -0.23077   0.30769
   0.23077   0.30769   0.92308

octave:3> v=[3;4;12]
v =

    3
    4
   12

octave:4> H*v
ans =

    0
    0
   13