I am looking for new ideas how to construct a guess for a (positive, hermitian) matrix A given some matrix-vector products Ax (with random vectors x). One such method would be to perform rank one updates to A each time a new Ax becomes available, and this is what is used in for example the BFGS update formula in quasi-Newton methods. However, this just updates a one dimensional subspace of the matrix. In my application it would be more natural to scale the whole matrix instead of performing a rank one update, but just a simple scaling is not enough because it cannot be consistent for all the Ax's. I have a rather good approximation to A to start with, so I know roughly the distributions of eigenvalues if that can help. My biggest problem is that I don't know how to even define the problem properly..
Perhaps I can sharpen the question with some feelback.
Edit: Learned about shrinkage estimation, perhaps that is the way to go.