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arXiv · 2609.05419

The 2M Multiplication Algorithm for Complex Matrices

Peter Caday
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Abstract

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.

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Peter Caday. 2026-05-15. The 2M Multiplication Algorithm for Complex Matrices. https://arxiv.org/abs/2609.05419

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