# What is the asymptotically fastest known algorithm for computing the nullspace of a matrix?

I know Gaussian Elimination takes $O(n^3)$ arithmetic operations, but I'm unsure if any better algorithms are known.

• I can see one way to do Gaussian Elimination of matrix M in O(n^2 rank(M)) time. Is there a way to do that faster? – Kyle Oct 24 '12 at 20:35

The exponent of computing a basis of the kernel is the same as the exponent of matrix multiplication, see the book Algebraic Complexity Theory by Bürgisser, Clausen & Shokrollahi. So it can be done in time $O(n^{2.38})$.