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 # The 2M Multiplication Algorithm for Complex Matrices

  ![](/sites/default/files/styles/wide/public/publications/Pic2M.png?itok=eJCO216E)

 Complex matrix multiplication is typically computed using 4 real matrix multiplications (GEMMs) of the same size. The well-known 3M multiplication algorithm reduces this cost to 3 real GEMMs, together with quadratic time pre- and post-processing steps. In this paper, we reduce 3M to 2M for matrices with integer real and imaginary parts, performing complex GEMM with only 2 real GEMMs of the same size, along with quadratic time pre- and post-processing. For floating-point matrices, 2M multiplication combines naturally with the Ozaki-II scheme, yielding a practical, high-performance algorithm for computing a complex floating-point GEMM in roughly twice the time of a real GEMM of the same size. As corollaries, we derive new algorithms for symmetric rank-*k* updates (SYRK/HERK) that internally use full rectangular GEMMs.

 ## Authors

Peter Caday (NVIDIA)

 ## Publication Date

Tuesday, June 30, 2026

 ## Research Area

[Algorithms and Numerical Methods](/index.php/research-area/algorithms)

[High Performance Computing](/index.php/research-area/high-performance-computing)

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