Question involving Householder Matrix This is an exam question that I'm having trouble solving.
Given a unit vector $v^Tv=1$, the Householder matrix is defined as $H=I-2vv^T$.
The first question is: given column vector $x$, if $Hx=c\cdot e_1$ where $c$ is constant and $e_1$ is the first vector of the canonical basis, find $c$.
After some algebra, I found that
$$c=x_1-2v_1\sum_{i=1}^n{v_ix_i}$$
I was unable to simplify it further.
The second question that I've been unable to solve is:
What is $v$, so that $H=I-2vv^T$ satisfies $Hx=c\cdot e_1$?

A Householder reflection is an isometric operation, thus $|c|=\|x\|$. 
To get more information, observe that as $Hx=ce_1$, so also $H(ce_1)=x$ and consequently 
$$
ce_1-2cvv_1=x\implies c-2cv_1^2=x_1, \;c=\frac{x_1}{1-2v_1^2}
$$
and thus
$$
x=\frac{(e_1-2vv_1)x_1}{1-2v_1^2}
$$
is up to a scaling factor completely determined by $v$.
This also is half the way to solve the second task.
