# The exponent of matrix multiplication

- kind: problem
- status: open
- year: 1969
- authors: after Volker Strassen's $O(n^{2.81})$ algorithm

**Statement.** Is $\omega = 2$, that is, can two $n \times n$ matrices be multiplied in $n^{2+\varepsilon}$ operations for every $\varepsilon > 0$?

Strassen broke the $n^3$ barrier, and the bound has crept down since, to about 2.371 by 2024. Each step now wins in the third decimal place. AlphaTensor in 2022 found new small-case algorithms by search, showing that machine search can contribute. A faster algorithm is a checkable object; a lower bound would need a proof.

Posed 1969, after Volker Strassen's $O(n^{2.81})$ algorithm.

## Sources

- [Wikipedia: Matrix multiplication algorithm](https://en.wikipedia.org/wiki/Matrix_multiplication_algorithm)

[Algorithms](/wiki/Space:algorithms.md) · [Computation](/wiki/Continent:computation.md)

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