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
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 $Hx=c\cdot e_1$?