Complex to Rational Fast Matrix Multiplication

Fast subcubic time matrix multiplication algorithms are asymptotically faster than the classical algorithm, but they are often slower in practice. One obstacle to speed is the use of complex coefficients, which increases arithmetic overhead and limits practical efficiency. This paper focuses on transforming complex-coefficient matrix multiplication algorithms into equivalent real or rational-coefficient ones. We present a method that, under some empirically mild condition, for complex-coefficient matrix multiplication algorithms, either proves that no equivalent rational algorithm exists or constructs a family of equivalent rational forms. Our approach relies on basic linear-algebraic properties of similarity transformations of complex matrices. This method recovers the results of Dumas, Pernet, and Sedoglavic (2025) and generalizes to other settings, including irrational to rational coefficients and rational to integer coefficients. Using this method, we show that no rational algorithm is equivalent to Smirnov's (4, 4, 9, 10) algorithm over (2022), that no real algorithm is equivalent to the (4, 4, 4, 48) algorithm of Kaporin over (2024), and prove the non-existence of integer coefficient form of a few algorithms.