Jtdylktuy

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 $mathbf{H} = e^{itheta}left(mathbf{I} - 2frac{mathbf{u}mathbf{u}^*}{mathbf{u}^*mathbf{u}}right)$, where $e^{itheta} = frac{overline{x_1}}{|x_1|}$ and $mathbf{u} = mathbf{x} - e^{itheta}|mathbf{x}|_2mathbf{e}_1$. 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 $mathbf{H}^* = mathbf{H^{-1}} = e^{-itheta}left(mathbf{I} - 2frac{mathbf{u}mathbf{u}^*}{mathbf{u}^*mathbf{u}}right)$ 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.

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 $Vsubseteqmathbb C^n$. For any $x,yinmathbb C^n$, if we write $x=u_x+v_x$ and $y=u_y+v_y$ where $u_x,u_yin V^perp$ and $v_x,v_yin V$, then
$$
langle Hx,yrangle
=langle u_x-v_x,u_y+v_yrangle
=langle u_x,u_yrangle - langle v_x,v_yrangle
=langle u_x+v_x,u_y-v_yrangle
=langle x,Hyrangle.
$$
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 $langle x,yrangle=langle x,Hxrangle$ must be real. In other words, you can construct a Householder reflection that maps $x$ to $y$ only when $langle x,yrangle$ is real.

This is an often neglected point. People usually only deal with Householder reflection over the reals. Since $langle x,yrangle$ 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 $langle x,e_1rangle=x_1$ to be real. If $x_1$ is not real, you cannot construct $H$. If $x_1$ is real instead, then $e^{itheta}=1$ 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.