Hermitian Property of a Householder Transform on a Complex Field Let $\mathbf{H}$ be a Householder (i.e. elementary reflector), such that $\mathbf{Hx} = \mathbf{e}_1$, for an $\mathbf{x} \in \Bbb{C}^n$, having $\|\mathbf{x}\|_2 = 1$.  
For this I have defined $\mathbf{H} = e^{i\theta}\left(\mathbf{I} - 2\frac{\mathbf{u}\mathbf{u}^*}{\mathbf{u}^*\mathbf{u}}\right)$, where $e^{i\theta} = \frac{\overline{x_1}}{|x_1|}$ and $\mathbf{u} = \mathbf{x} - e^{i\theta}\|\mathbf{x}\|_2\mathbf{e}_1$.  I am indeed getting the expected result of $\mathbf{Hx} = \mathbf{e}_1$, but my trouble is figuring out how to enforce the property that $\mathbf{H} = \mathbf{H}^*$.  Can anyone please point me in the right direction?
I have also checked that this $\mathbf{H}$ is unitary, since $\mathbf{H}^* = \mathbf{H^{-1}} = e^{-i\theta}\left(\mathbf{I} - 2\frac{\mathbf{u}\mathbf{u}^*}{\mathbf{u}^*\mathbf{u}}\right)$ and $\mathbf{H}^*\mathbf{H} = \mathbf{I}$, but again, I don't see how $\mathbf{H} = \mathbf{H^{-1}}$.  I even get that $\mathbf{H}^{-1}\mathbf{e}_1 = \mathbf{x}$, which should be expected.
Any help is really well appreciated. 
 A: It is not always possible to construct a Householder reflection that maps a prespecified unit vector in $\mathbb C^n$ to another.
A Householder reflection $H$, by definition, is a linear operator whose restrictions are $-\operatorname{Id}$ on $V$ and $\operatorname{Id}$ on $V^\perp$ for some one-dimensional subspace $V\subseteq\mathbb C^n$. For any $x,y\in\mathbb C^n$, if we write $x=u_x+v_x$ and $y=u_y+v_y$ where $u_x,u_y\in V^\perp$ and $v_x,v_y\in V$, then
$$
\langle Hx,y\rangle
=\langle u_x-v_x,u_y+v_y\rangle
=\langle u_x,u_y\rangle - \langle v_x,v_y\rangle
=\langle u_x+v_x,u_y-v_y\rangle
=\langle x,Hy\rangle.
$$
It follows that $H$ is necessarily Hermitian. Being Hermitian is not an optional property, but a must.
Consequently, if $y=Hx$ for some unit vectors $x$ and $y$, then $\langle x,y\rangle=\langle x,Hx\rangle$ must be real. In other words, you can construct a Householder reflection that maps $x$ to $y$ only when $\langle x,y\rangle$ is real.
This is an often neglected point. People usually only deal with Householder reflection over the reals. Since $\langle x,y\rangle$ is real in this case, they don't realise that the story is a bit different over $\mathbb C$.
In your case, if you want $Hx=e_1$, we need $\langle x,e_1\rangle=x_1$ to be real. If $x_1$ is not real, you cannot construct $H$. If $x_1$ is real instead, then $e^{i\theta}=1$ and all apparent difficulties in your question vanish. The construction is then analogous to that over $\mathbb R$. Just replace any matrix/vector transpose by conjugate transpose.
Some people do try to generalise Householder reflection in the real case to the complex case in a similar manner to your question. Now the construction is always possible. However, the resulting $H$ in general is neither Hermitian nor a reflection, but a complex scalar multiple of a Hermitian Householder reflection.
