Lower Bound on the Cost of Solving Linear System The cost of solving a linear system ("exactly") with Gauss Elimination and other similar methods with a few right hand side and where the matrix has no structure is $\mathcal{O}(N^3)$ where $N$ is the system size.
I am wondering about the lower bound for solving a linear system. An obvious lower bound is $\mathcal{\Omega}(N^2)$ (since the information content is $\mathcal{O}(N^2)$). Are there better lower bounds other than $\mathcal{\Omega}(N^2)$ for solving the linear system? Is there a way to prove that the lower bound of $\mathcal{\Omega}(N^2)$ can never be hit for a matrix with no special structure? (assume that we are solving a system with only one/few right hand side).
Also are there other algorithm which solve these system "exactly" whose cost is less than $\mathcal{O}(N^3)$?  I am aware of Strassen algorithm which perform matrix multiplications in $\mathcal{O}(N^{\log_27})$. I assume this can be used to solve a linear system in $\mathcal{O}(N^{\log_27})$. (?)
(The system has no special structure. I am not worried about the stability and other numerical intricacies of the method as of now. I would appreciate if someone could point to some work done in this regard.)
Thanks
 A: Let me take this opportunity to make a comment that talks about the upper bound, not the lower.
Actually, let me remind you that it is a common false belief that Gaussian Elimination has $O(n^3)$ complexity. See the nice question with its answers here at cstheory
This misbelief happens because even though GE requires $O(n^3)$ arithmetic operations, if not done properly, there can be massive intermediate coefficient growth, which renders judging the true complexity of GE a difficult task.
You might also enjoy looking here at the near linear time algorithms for solving special linear systems (example for diagonally dominant matrices). For general, non-structured matrices, the situation is less clear.
A: Any $O(N^t)$ algorithm for matrix multiplication yields a corresponding $O(N^t)$ algorithm for matrix inversion. There are not any non-trivial lower bounds for matrix multiplication. It is believed that there exist $O(N^{2 + \epsilon})$ algorithms for any $\epsilon > 0$, but as far as I know there are not believed to be $O(N^2)$ algorithms.
