Let $v\in\Bbb{C}^{n}$ be a fixed vector. Then does there exist a unitary Matrix $U$ such that $Uv=\|v\|e_{1}$? or even $Uv=c\|v\|e_{1}$ for some real constant $c$?
I am looking for a Householder reflection sort of a transformation.
For example, if $v\in\Bbb{R}^{n}$ or even if $\langle v,e_{1}\rangle\in\Bbb{R}$, then the standard Householder reflection $H=I-2uu^{*}$ where $u=\frac{v-\|v\|e_{1}}{\|v-\|v\|e_{1}\|}$ would suffice. But this won't work for general $v\in\Bbb{C}^{n}$.
If $v=0$, any unitary matrix works. Let's then assume $v \neq 0$, and let $f_1 = v/\|v\|$, and complete it into an Hermitian basis $\{f_1,\ldots,f_n\}$. Consider the $\Bbb C$-linear map $u\colon \Bbb C^n \to \Bbb C^n$ defined by its action on the basis $\{f_1,\ldots,f_n\}$ as $u(f_i) = e_i$, with $\{e_1,\ldots,e_n\}$ the canonical basis. Then $u$ is unitary: indeed, if $x = \sum_{i=1}^n\lambda_if_i$, then $$ \|u(x)\|^2 = \left\|\sum_{i=1}^n \lambda_i e_i\right\|^2 = \sum_{i=0}^n|\lambda_i|^2 = \left\|\sum_{i=0}^n \lambda_i f_i\right\|^2 = \|x\|^2. $$ This should solve your problem.