I want to find the complex vector $x$ of length $N$ which solves
$$ z = \text{diag}(M x) x^*$$
where the complex vector $z$ of size $N$ as well as the complex, non-singular $N\times N$ matrix M are given. Note that $x^*$ is the complex conjugate of $x$.
Alternatively, the diagonalized vector may be shifted by a real scalar value $\lambda$ such that
$$ z = \text{diag}(M x + \lambda) x^*$$
I imagine the solution $(x)$ or $(x, \lambda)$ to be degenerate. In this case, the question would be what is one possible solution, ie the principal value of the solution?