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?



Bound on smallest entry of inverse matrix
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*](http://imajna.oxfordjournals.org/content/12/4/463.full.pdf)


[Golub & Meurant, *Matrices, Moments and Quadrature*](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.56.6363)


[Benzi & Golub, *Bounds for the entries of matrix functions with applications to preconditioning*](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.134.5084) 


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](https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&ved=0CCgQFjAA&url=http://www.math.technion.ac.il/iic/ela/ela-articles/articles/vol20_pp291-302.pdf&ei=cMQPU6LcLpGihgeU9oH4DQ&usg=AFQjCNHTJkCq-WaEhEdJ6aiGv3A7kLTVjQ&sig2=Q4dHAXrgFx9MlSN_7jVceA&bvm=bv.61965928,d.ZG4).


I had studied this subject in some detail so if you are interested, I can try to help more specifically as well.


