If I know y (samples) and x (signal), is there a practical matlab algorithm to calculate A in y = Ax?
You could use tensor factorization methods. For example if you are sure that your data is non-negative, then Nonnegative Matrix Factorization might be used.
Depending on your application and format of A, you might want the sparsest solution. In that case minimization of $L_0$ norms would be considerable.