I have seen in several places that the inverse of a sparse matrix is generally not sparse, but I have failed to find more in-depth analysis than empirical or case-by-case studies.
My question is the following : is there a general way to characterize sparse symmetric positive-definite matrices whose inverses are also sparse ? Can we extend from the trivial case of diagonal matrices, for example, a way to say that the inverse of a "very" sparse symmetric PD matrix is sparse too ?
I realize that the definition of what constitutes a "sparse" matrix and its "sparse" inverse is always quite shaky. Let's say that we define an inverse $A^{-1}$ to be a "sparse inverse" if it doesn't have "many more" non-zero elements than $A$. So, if the ratio of the number of null elements of $A$ and the number of null elements of $A^{-1}$ is not "exploding" or close to null.
Some trivial cases are diagonal matrices or orthogonal sparse matrices.
