Rank of the 3×3 matrix multiplication tensor

Determine the exact tensor rank of the trilinear matrix multiplication tensor for multiplying 3×3 matrices.

Background

To design fast algorithms via Schönhage’s asymptotic sum inequality, one ideally bounds the ranks of matrix multiplication tensors. Direct rank bounds, however, are difficult. As a canonical example, the exact rank of the 3×3 matrix multiplication tensor remains unresolved, which is why modern approaches rely on indirect methods such as the laser method applied to structured tensors like CW_q.

This open problem underscores a core obstacle behind rank-based algorithm design and motivates the indirect strategies developed and extended by the authors.

References

However, directly bounding the rank of a matrix multiplication tensor is quite difficult (for example, even determining the rank of the tensor for multiplying 3 × 3 matrices is still an open problem today), and so algorithms since the work of Coppersmith and Winograd [cw90] have used an indirect approach:

— More Asymmetry Yields Faster Matrix Multiplication  (2404.16349 - Alman et al., 2024) in Section 2.1 (Technical Overview: The Laser Method and Asymptotic Sum Inequality)

This proof leaves open the question of whether or not a 22-multiplication algorithm is possible for $3\times3$ matrix multiplication.

— Lower Bound of 22 for 3x3 Matrix Multiplication over the Integers  (2610.01639 - Rudich et al., 1 Oct 2026) in Section Conclusion

The rank of $$ over $\mathbb{F}_3$, over $\mathbb{Q}$ and over $\mathbb{C}$ remains open, as does its border rank over $F$.

— The rank of $3\times 3$ matrix multiplication over $\mathbb{F}_2$ is 23  (2610.11823 - Tomar, 8 Oct 2026) in Section 'Remarks'