New ways to multiply 3 x 3-matrices.

Abstract It is known since the 1970s that no more than 23 multiplications are required for computing the product of two 3 × 3 -matrices. For non-commutative coefficient rings, it is not known whether it can also be done with fewer multiplications. However, there are several mutually inequivalent ways of doing the job with 23 multiplications. In this article, we extend this list considerably by providing more than 17,000 new and mutually inequivalent schemes for multiplying 3 × 3 -matrices using 23 multiplications. Moreover, we show that the set of all these schemes is a manifold of dimension at least 17.