Designing Strassen's algorithm

TL;DR

This paper provides a clear, simple proof of Strassen's matrix multiplication algorithm using 2-designs and group theory, and extends the construction to all matrix sizes.

Contribution

It offers the simplest proof of Strassen's algorithm and generalizes the construction to all matrix sizes using group orbit 2-designs.

Findings

Simplified proof of Strassen's algorithm using unitary 2-designs.

Generalization of the algorithm to all matrix sizes via group orbits.

The resulting algorithms are not optimal for sizes greater than 3.

Abstract

In 1969, Strassen shocked the world by showing that two n x n matrices could be multiplied in time asymptotically less than $O(n^3)$. While the recursive construction in his algorithm is very clear, the key gain was made by showing that 2 x 2 matrix multiplication could be performed with only 7 multiplications instead of 8. The latter construction was arrived at by a process of elimination and appears to come out of thin air. Here, we give the simplest and most transparent proof of Strassen's algorithm that we are aware of, using only a simple unitary 2-design and a few easy lines of calculation. Moreover, using basic facts from the representation theory of finite groups, we use 2-designs coming from group orbits to generalize our construction to all n (although the resulting algorithms aren't optimal for n at least 3).