Let a square $n\times n$ real matrix ${\bf A}$ and two vectors ${\bf x}$ and ${\bf b}$ of length $n$, such that $${\bf A}{\bf x}={\bf b}.$$ Solving for ${\bf x}$ through standard Gaussian Elimination yields an aggregate complexity of almost $O(n^3)$. However, there are cases where solving (or $\epsilon$-approximately solving) for ${\bf x}$ costs $O(n\log^\rho n)$, such as systems where ${\bf A}$ is a symmetric and diagonally dominant matrix (e.g., a Laplacian) [1].

Which other families of linear systems (i.e., matrices) admit linear (or nontrivial poly(n)) time solutions? If we consider finite fields instead of real matrices, are there any families of matrices there that admit nearly linear time solutions?

[1] http://www.cs.yale.edu/homes/spielman/Research/linsolve.html