# Commutative matrix multiplication algorithms

What is known about commutative algorithms like Winograd algorithm and its variants for Matrix Multiplication? Why is there not much study on them? Can they be asymptotically as efficient as Non-commutative algorithms?

Update

Existence of $O(n^\omega)$ non-commutative algorithms for matrix multiplication implies existence of $O(n^\omega)$ commutative algorithms. Does converse statement hold truth?

In answer to the "Update": yes, for any $c$, the existence of an $O(n^c)$ non-commutative algorithm for matrix multiplication is equivalent to the existence of an $O(n^c)$ commutative algorithm for matrix multiplication (I am assuming, here, that all algorithms are algebraic, in the sense of algebraic circuits). This is because any algebraic circuit computing a bilinear map can be converted into a bilinear circuit at the expense of at most a factor of 2 in the size.