Multiplying two matrices is a basic computational task.
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The AlphaEvolve report gives a rank-48 decomposition of the 4×4 matrix-multiplication tensor over the complex numbers, yielding a bilinear algorithm using 48 scalar multiplications.
Read original source AlphaEvolve technical report, §3.1arxiv.org
Original report: 2025-05-14Dupont and collaborators report ω<2.371177 by improving the combination-loss optimization and refining it with AlphaEvolve.
Original report: 2026-08-17The manuscript reports that the complex matrix-multiplication exponent is at most 9/4.
Read original source Original manuscript · family 107github.com
Original report: 2026-10-06The authors report an algorithm multiplying 4×4 matrices over the two-element field using 47 scalar multiplications, compared with 49 from two levels of Strassen’s algorithm.
Original report: 2022-10-05Multiplying two matrices is a basic computational task. Different algorithms trade arithmetic operations, numerical stability, and actual running time. A small-matrix improvement is different from an improvement to the asymptotic exponent.
The familiar algorithm computes each output entry as a dot product. Strassen showed that arithmetic operations can be saved by reorganizing this computation. Tensor decompositions describe multiplication algorithms by expressing the operation as a sum of simpler products.
There are three separate targets: fewer scalar multiplications at fixed dimensions, a better asymptotic exponent as matrices grow, and faster execution on real hardware. A result can advance one without advancing the others. The field matters: an identity over the two-element field need not give the same algorithm over the real or complex numbers.
Background review: 2026-10-11. Selected sources:
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