For a symmetric, invertible matrix $A=(a_{ij})\in \mathbb{R}^{n\times n}$ with (at least two) nonzero off-diagonal elements, is it possible to bound in absolute value the smallest entry of its inverse in terms of the $a_{ij}$? And if so how?
If this is difficult to answer in such a general setting, would assuming that $A$ is either positive definite or sparse help simplifying the problem? Are there other assumptions that could help simplifying the problem?
Best Answer
Well, you are asking for a lot, but with some assumptions there are results for this problem. There is also a big difference between diagonal entries and off-diagonal and lower and upper bounds. But it's not hopeless!
Have a look at these papers:
Robinson & Wathen, Variational bounds on the entries of the inverse of a matrix
Golub & Meurant, Matrices, Moments and Quadrature
Benzi & Golub, Bounds for the entries of matrix functions with applications to preconditioning
These papers mostly handle the positive definite case. Another kind of approach, which doesn't even require symmetry, assumes instead diagonal dominance of $A$. This kind of stuff is mostly embedded as lemmas in other things so one has to know where to look for it. For example, Lemma 2.1 here.
I had studied this subject in some detail so if you are interested, I can try to help more specifically as well.