Hadamard product with rank 1 matrix Let $\odot$ represent Hadamard or pointwise multiplication. If $\pmb{Y}$ is a $\underline{given}$ positive definite square matrix, can i estimate a positive semi-definite $\pmb{X}$ matrix such that 
\begin{equation}
\pmb{Y} = \pmb{X} \odot \pmb{z}\pmb{z}^H
\end{equation} 
where $\pmb{z}$ is a complex vector with phases only (i.e. each element of $\pmb{z}$ is of magnitude $1$). Also $(.)^H$ stands for conjuate-transposition, or Hermitian (Thanks @igael). I really don't know where to start. Maybe, i have formulated a problem that does not make sense.
$\mathbf Y = diag(\mathbf z)\mathbf X diag(\mathbf z^H)$. Since $diag(\mathbf z)diag(\mathbf z^H)=\mathbf I$, $\mathbf X$ can be estimated as $\mathbf X = diag(\mathbf z^H)\mathbf Y diag(\mathbf z)$, where $\mathbf z$ is any vector with your constraint.
$\mathbf Y = diag(\mathbf z)\mathbf X diag(\mathbf z^H)$. Since $diag(\mathbf z)diag(\mathbf z^H)=\mathbf I$, $\mathbf X$ can be estimated as $\mathbf X = diag(\mathbf z^H)\mathbf Y diag(\mathbf z)$, where $\mathbf z$ is any vector with your constraint.