Is the matrix multiplication exponent equal to 2?

Still open as of 2026.

In 1969 Volker Strassen showed that two $2\times 2$ matrices can be multiplied with 7 multiplications instead of 8, and applying this recursively gives an $\mathcal{O}(n^{2.8074})$ algorithm. That surprise started a long sequence of improvements through increasingly sophisticated tensor methods. In 2024 Alman, Duan, Vassilevska Williams, Xu, Xu and Zhou brought the bound to about 2.3713, and a further small improvement to about 2.3712 was reported in 2026. Many researchers conjecture that $\omega =2$, but no lower bound beyond the trivial $\omega \ge 2$ is known for general algorithms.

References

• Computational complexity of matrix multiplication (Wikipedia)
• J. Alman et al., More Asymmetry Yields Faster Matrix Multiplication (arXiv:2404.16349)