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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.

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I see that this question has been inactive for a very long time, but I find the question interesting. When you say you know $Ax$ for some random $x$, do you mean you know both $x$ and $Ax$, or do you only know $Ax$ and the distribution of $x$? – Mårten W Sep 12 '13 at 22:05
Can you give some more details. You say you have some information about $A$, that might be expressed as a prior distribution in a bayes analysis, and the random matrix-vector products would be the data used to update the prior to give the posterior. Or maybe methods from tomography? – kjetil b halvorsen Apr 9 '14 at 19:50

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