matrix multiplication algorithm time complexity

Question

I came up with this algorithm for matrix multiplication. I read somewhere that matrix multiplication has a time complexity of o(n^2). But I think my this algorithm will give o(n^3). I don't know how to calculate time complexity of nested loops. So please correct me.

for i=1 to n    for j=1 to n          c[i][j]=0      for k=1 to n          c[i][j] = c[i][j]+a[i][k]*b[k][j] 

Answer 1

Using linear algebra, there exist algorithms that achieve better complexity than the naive O(n³). Solvay Strassen algorithm achieves a complexity of O(n^2.807) by reducing the number of multiplications required for each 2x2 sub-matrix from 8 to 7.

The fastest known matrix multiplication algorithm is Coppersmith-Winograd algorithm with a complexity of O(n^2.3737). Unless the matrix is huge, these algorithms do not result in a vast difference in computation time. In practice, it is easier and faster to use parallel algorithms for matrix multiplication.

Answer 2

The naive algorithm, which is what you've got once you correct it as noted in comments, is O(n^3).

There do exist algorithms that reduce this somewhat, but you're not likely to find an O(n^2) implementation. I believe the question of the most efficient implementation is still open.

See this wikipedia article on Matrix Multiplication for more information.