 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:and beating the Strassen algorithm using 3x3 matrices would require a non-commutative algorithm with 21 multiplications.

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