= Commutative matrix multiplication algorithm
A "commutative matrix multiplication algorithm" is a matrix multiplication algorithm that requires the ring to be commutative. Such algorithms are inferior because you cannot use them to create more efficient algorithms for <general matrix matrix multiplication> by decomposing the bigger matrix into smaller ones.
For example, the <Strassen algorithm> is based on reduction to non-commutative <2x2 matrix multiplication> optimized to be done in 7 multiplications rather than 8 as in the native algorithm.
For <3x3 matrix multiplication>, the best algorithms as of 2025 are:
* commutative: 21 multiplications
* non-commutative: 23 multiplications
and beating the <Strassen algorithm> using 3x3 matrices would require a non-commutative algorithm with 21 multiplications.
Bibliography:
* https://stackoverflow.com/questions/10827209/ladermans-3x3-matrix-multiplication-with-only-23-multiplications-is-it-worth-i
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