I have some experience with non-linear least squares curve fitting. For instance, if I want to fit a Gaussian curve to a set of data, I would use a non-linear least squares technique. A "model" matrix is implemented and combined with the observed data. The solution is found by applying well defined methods of Linear Algebra. Here is a link to a non-linear least squares method I have implemented many times in the past. http://mathworld.wolfram.com/NonlinearLeastSquaresFitting.html I bring this up because I am trying to understand how Matrix Mechanics is used in quantum physics. The above curve-fitting example is the closest I can come, from my own experience, to understanding what might be going on. Does matrix mechanics depend on the observed results? Is Matrix Mechanics, as used in the quantum world, fitting the observed results to a "model" matrix of some kind? (density/scatter matrix?) And the eigenvalues of the combined system(observed results + model matrix) are the solutions? Is this a possible analogy? I am not looking to be right, I am just trying to figure out how Matrix Mechanics is actually used. Does it operate on a "wave function". And if so, where does that "wave-function" come from? It is my understanding that Matrix Mechanics and Schrodinger Equation are completely different , but equivalent, methods, but , to my understanding, only the Schrodinger Equation makes use of a wave equation.