OP-00067 Open

Border rank of the 3×3 matrix product

Let $M_3 = \sum_{i,j,k=1}^{3} e_{ij} \otimes e_{jk} \otimes e_{ki} \in \mathbb{C}^9 \otimes \mathbb{C}^9 \otimes \mathbb{C}^9$ be the tensor of $3 \times 3$ matrix multiplication, and let $\underline{R}$ denote border rank over $\mathbb{C}$. Conner, Harper and Landsberg proved $\underline{R}(M_3) \ge 17$, and Smirnov showed $\underline{R}(M_3) \le 20$.

Problem. Determine $\underline{R}(M_3)$.

Source: Classical, with no single originator; Strassen (1983) proved the first lower bound, 14. See Conner, Harper and Landsberg (2023) and Alman and Li (2026).

Imported from github.com/ajt60gaibb/OpenProblemsInNLA.

Classification

Primary: AG — Algebraic Geometry
Secondary: NA
Keywords: border rank, matrix multiplication, OpenProblemsInNLA, tensor rank
Literature checked: Oct. 2, 2026 by clemens-admin

References

@article{strassen1983,
  author  = {Strassen, Volker},
  title   = {Rank and optimal computation of generic tensors},
  journal = {Linear Algebra Appl.},
  volume  = {52/53},
  year    = {1983},
  pages   = {645--685},
}
@article{conner2023,
  author        = {Conner, Austin and Harper, Alicia and Landsberg, J. M.},
  title         = {New lower bounds for matrix multiplication and $\operatorname{det}_3$},
  journal       = {Forum Math. Pi},
  volume        = {11},
  year          = {2023},
  pages         = {e17},
  doi           = {10.1017/fmp.2023.14},
  eprint        = {1911.07981},
  archivePrefix = {arXiv},
}
@article{smirnov2013,
  author  = {Smirnov, A. V.},
  title   = {The bilinear complexity and practical algorithms for matrix multiplication},
  journal = {Comput. Math. Math. Phys.},
  volume  = {53},
  number  = {12},
  year    = {2013},
  pages   = {1781--1795},
  doi     = {10.1134/S0965542513120129},
}
@misc{alman2026,
  author        = {Alman, Josh and Li, Baitian},
  title         = {Asymptotic Rank Speedup Theorems, Revisited},
  year          = {2026},
  eprint        = {2605.21738},
  archivePrefix = {arXiv},
}
@misc{openproblemsinnla-ac03,
  author       = {Townsend, Alex},
  title        = {{AC-03}: Border rank of the $3\times3$ matrix product},
  howpublished = {Open Problems in Numerical Linear Algebra, GitHub repository},
  year         = {2026},
  url          = {https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/main/arithmetic-and-complexity/AC-03/README.md},
  note         = {Read at commit e375a6f, 2026-09-30},
}

Citing this problem

This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.

OP-00067, "Border rank of the 3×3 matrix product", clopen.solutions.