The essence of my question is, if I have a Hermitian matrix that is linearly dependent on a set of parameters and I have an estimate of its eigenvalues, is there a "simple" way to determine the values of the parameters? Ideally, I would also like to have some measure of the goodness of the fit and the degree of variation within the parameters.
As a materials physicist, I often have to create a simple quantum mechanical model from either experimental data or a more complex calculation. For the smaller problems (8x8, with 10 params), the parameters can be found by painstakingly working through the various relationships among the parameters, due to symmetry, etc. But, this method is specific to each problem and does not scale well to larger problems. For instance, one system I'm looking at would require a 20x20 matrix with 21 parameters, and that is without including spin! Alternatively, there is the brute force method of simulated annealing, which involves taking a random walk through the parameter space, and slowly decreasing the step size in the hopes that the calculation will get stuck in the global minimum. Neither of these methods is particularly appealing, so I'd like some ideas on how to approach this in a consistent manner.