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

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  • $\begingroup$ 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? $\endgroup$ – Kyle Oct 24 '12 at 20:35
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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 $O(n^{2.38})$.

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    $\begingroup$ or 2.372 now, right ? $\endgroup$ – Suresh Venkat Oct 12 '12 at 13:41
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    $\begingroup$ I think it's 2.3727. $\endgroup$ – Markus Bläser Oct 12 '12 at 14:52

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