Modify the Householder reflector so that it works for a vector with complex entries
Sebastian Wright 
Explain how to modify the definition of the Householder reflector so that it works for a vector with complex entries. That is, given $x \in \mathbb C^m$ , how should be define the unit vector $v$ so that $(I_d−2vv^*)x = \|x\|e_1$?
I know $\mathbb C^m$ is an actually 2m dimensional space, so how to modify the above formula in order to suit for $\mathbb C^m$ case?
$\endgroup$2 Answers
$\begingroup$The modification goes as follows \begin{align} \hat{u} &= \frac{x + e^{i\theta} ||x||e_1}{||x + e^{i\theta} ||x||e_1||}, \\ H &= -e^{-i\theta}(I - 2 \hat{u} \hat{u}^*), \end{align} where $x_1=e^{i\theta}|x_1|$.
In this case, you have $Hx = ||x|| e_1$.
See at page 19 for more details.
$\endgroup$ $\begingroup$This is discussed in lots of textbooks on numerical linear algebra. See for example:
$\endgroup$ 4
