# Can Strassen's multiplication algorithm be improved if we divide matrices to 3x3 or axa in general?

Strassen's algorithm uses the divide and conquer approach to divide the matrix multiplication of two nxn matrices to multiplication of 7 2x2 matrices to get an overall complexity O(n^c) where c=log_2(7). I also read a paper which proved that it was impossible to multiply two 2x2 matrices in less than 7 multiplication operations.

So my question is, what's so special about 2x2? Why not break the original matrix multiplication to 3x3 matrix multiplication. Surely, if we can make the multiplication in 21 multiplications or less, then we would have a better result than dividing it to 2x2. And then why stop at 3x3, when you can divide the problem to 4x4, or in general axa?

• You'd need to calculate 9 sums from 27 products, but find 21 clever products of sums instead so that these 9 sums can be produced as sums/differences of those 21 products. That seems difficult. You have to save 6 of 27 products. Nothing keeps you from trying. – gnasher729 Feb 18 '17 at 15:56

Suppose that the algebraic complexity of matrix multiplication is $O(n^\alpha)$. Then for every $\epsilon > 0$ there is a divide-and-conquer algorithm for multiplying matrices in $O(n^{\alpha+\epsilon})$.
Here a divide-and-conquer algorithm is an algorithm which recursively applies some smart algorithm for multiplying $m\times m$ matrices, for some constant $m$.