[abeppu]

Predicting that this holds for any j > 0 seems rather bold. Would you care to share your intuition why you think that's the case?

[roflmaostc]

Two matrices with size NxN each can be multiplied naively with the schoolbook algorithm in O(N^3).

It's clear that the algorithm needs at least O(N^2) because to access each element of the matrices once, you need a double for loop, which is O(N^2).

  for i in rows
      for j in cols
          # do something with element matrix1 [i, j], matrix2[i, j],...

so it has to be j >= 0

[JohnKemeny]

His question was: what is the reasoning behind there existing an algorithm running in time n^2+epsilon for really small epsilon.

[abeppu]

Yeah, we're agreed that j cannot be less than 0. But your conjecture was about _every j > 0_. Do you have any specific line of reasoning which suggests that j can be arbitrarily close to 0 (0 is the greatest lower bound)? Why do you not think there's some other specific limit k \in (0, 0.3728596] beyond which j cannot be improved?