I have a semi unitary matrix $A_{i,j}$ with $1 \leq j \leq N$, $1 \leq i \leq M$ and $M\geq N$, i.e. $A^\dagger A = I$. I now have a set $N$ equations for the squared entries of each row: $$\sum_j |A_{i,j}|^2 = n_i.$$ Is there a unique solution for this equation and if so, how do I find it? If it is not unique, can I construct at least one solution?
Or stated differently: if I only now that $$\sum_j |A_{i,j}|^2 = n_i,$$ can I create a semi-unitary matrix from this?