OP-00066 Partially resolved

Is the matrix multiplication exponent two?

Let $\omega$ be the exponent of matrix multiplication over $\mathbb{C}$: the infimum of the $\tau$ for which some arithmetic straight-line program, using additions, subtractions, multiplications and divisions and defined on every input, computes $AB$ for all $A, B \in \mathbb{C}^{n \times n}$ with $O(n^\tau)$ operations. Dupont et al. (2026) report the computer-assisted bound $\omega < 2.371177$; the only known lower bound is the trivial $\omega \ge 2$.

Problem. Is $\omega = 2$? Equivalently, can the product of two $n \times n$ complex matrices be computed with $O(n^{2+\varepsilon})$ arithmetic operations for every $\varepsilon > 0$?

Source: Classical, open since Strassen (1969) showed ω ≤ log₂ 7; see Bläser, Fast Matrix Multiplication (2013).

Imported from github.com/ajt60gaibb/OpenProblemsInNLA.

Imported from github.com/MColbrook/AIM.

Classification

Primary: NA — Numerical Analysis
Secondary: AG
Keywords: arithmetic complexity, matrix multiplication, OpenProblemsInNLA
Literature checked: Oct. 2, 2026 by clemens-admin

References

@article{strassen1969,
  author  = {Strassen, Volker},
  title   = {Gaussian elimination is not optimal},
  journal = {Numer. Math.},
  volume  = {13},
  number  = {4},
  year    = {1969},
  pages   = {354--356},
  doi     = {10.1007/BF02165411},
}
@book{blaser2013,
  author    = {Bl{\"a}ser, Markus},
  title     = {Fast Matrix Multiplication},
  series    = {Theory of Computing Library, Graduate Surveys},
  number    = {5},
  publisher = {Theory of Computing Library},
  year      = {2013},
  pages     = {1--60},
  doi       = {10.4086/toc.gs.2013.005},
}
@misc{dupont2026,
  author        = {Dupont, Emilien and Eisenberger, Marvin and Kozlovskii, Borislav and Mehrabian, Abbas and Ruiz, Francisco J. R. and See, Abigail and Zhou, Renfei and Alman, Josh and Vassilevska Williams, Virginia and Balog, Matej},
  title         = {Improving the matrix multiplication exponent with modern optimization and {AlphaEvolve}},
  year          = {2026},
  eprint        = {2608.16884},
  archivePrefix = {arXiv},
}
@misc{alman2026,
  author        = {Alman, Josh and Li, Baitian},
  title         = {Asymptotic Rank Speedup Theorems, Revisited},
  year          = {2026},
  eprint        = {2605.21738},
  archivePrefix = {arXiv},
}
@misc{openproblemsinnla-ac01,
  author       = {Townsend, Alex},
  title        = {{AC-01}: Is the matrix multiplication exponent two?},
  howpublished = {Open Problems in Numerical Linear Algebra, GitHub repository},
  year         = {2026},
  url          = {https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/main/arithmetic-and-complexity/AC-01/README.md},
  note         = {Read at commit e375a6f, 2026-09-30},
}

Resolution claims

An Upper Bound of 9/4 for the Matrix Multiplication Exponent v1 GitHub ↗

OpenAI

We prove that the exponent of matrix multiplication over the complex numbers is at most $9/4$.

Partial resolution · not verified · fidelity not established · posted on GitHub 6 Oct 2026

Status history

When	Change	By	Reason
2026-10-09	Open → Partially resolved	The registry's curators	An Upper Bound of 9/4 for the Matrix Multiplication Exponent v1 recorded from math @ fd4aeeb, build/paper.tex.

Citing this problem

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

OP-00066, "Is the matrix multiplication exponent two?", clopen.solutions.