Reconstructing operator matrix from subspace samples

[uekstrom]

Hi,
I wonder if there is some agreed-upon best way to reconstruct the matrix of a positive definite operator A using "sampling" (like in tomography). More in detail I want to do this:

I have many small sets of basis functions. The sets are in general not orthogonal. I compute matrix elements <i|A|j>, where |i> and |j> belong to the same "set". In other words, in the non-orthogonal basis I know certain diagonal blocks of A, while the other elements are unknown. I want to determine an estimate of the off diagonal elements.

One way of reconstructing A is to simply take any orthogonal basis for the union of all basis functions, and then work with that. However, the orthogonal basis is not unique. My question is if there is a best way of doing this?

 

[fresh_42]

You don't need orthogonality, linear independence is sufficient. And it gives you an entire linear equation system for the unknown $A_{ij}$ not only the diagonal. The question is alone which part of the domain you can cover by your small sets.