I am trying to solve $\mathbf{Ax}=\mathbf{b}$, but this time $\mathbf{A}$ is unknown. I do, however, have a pretty good starting guess for $\mathbf{A}$, but it needs some slight modifications. I know the problem is very underdetermined, but is there a way/method of estimating $\mathbf{A}$ based on the initial guess? $\mathbf{x}$ and $\mathbf{b}$ contain noise, so an exact solution does not exist.
Even just some keywords to help me in my further search would be very much appreciated.