Let A be a known complex matrix, B a complex vector, and C the complex vector to be solved. Imagine that we know that AC = B . Let assume that the number of lines of A and B are as many as needed. (In my problem, with an unchanged size for C, I can easily increase the number of lines of A and B ). How can I find C when only Q = [|B1|, |B2|, |B3], |B4|, ...,|Bn|] is known ?
So far I tried to solve this problem by multiplying by the conjugate : (AC)*conj(AC) = Q*Q, where * is element by element multiplication. but at the end, i didn't find any solution.
Too long for a comment; to recapitulate: Given are a matrix $A\in \Bbb C^{m\times n}$ with complex components and a vector $Q\in\Bbb R^{m\times 1}$ of non-negative real numbers. Further, the number $m$ of rows of $A$ and $Q$ is at least twice the number $n$ of columns of $A$, $m\ge 2n$.
We now consider the torus $$ T=\{B\in \Bbb C^m: |B_k|=Q_k\} $$ and ask for solutions $C$ of $$ AC\in T. $$
Geometrically, this means to compute the intersection of the hypertorus $T$ with real dimension $m$ (as long as all components of $Q$ are positive) and the subspace $Im(A)$ of real dimension $2n$. Since the real dimension of the full space is $2m$, and $2n+m\le 2m$, chances are high that the intersection is empty.
Put another way, if $a_k$ are the rows of $A$, then the vector $C$ with $2n$ real unknowns is bound by $m\ge 2n$ non-linear equations $|a_kC|^2=Q_k^2$, thus the non-linear system of equations is over-determined.