Complex to Rational Fast Matrix Multiplication

TL;DR

This paper introduces a systematic linear-algebraic method to convert complex matrix multiplication algorithms into rational or real schemes, addressing practical efficiency issues and proving non-existence of certain schemes.

Contribution

It provides a general framework for transforming complex algorithms into rational or real ones and extends previous ad hoc results to broader settings.

Findings

No rational scheme exists for Smirnov's algorithm.

No real scheme exists for Kaporin's complex algorithm.

Framework can prove non-existence of integer-coefficient schemes.

Abstract

Fast matrix multiplication algorithms are asymptotically faster than the classical cubic-time algorithm, but they are often slower in practice. One important obstacle is the use of complex coefficients, which increases arithmetic overhead and limits practical efficiency. This paper focuses on transforming complex-coefficient matrix multiplication schemes into equivalent real- or rational-coefficient ones. We present a systematic method that, given a complex-coefficient scheme, either constructs a family of equivalent rational algorithms or proves that no equivalent rational scheme exists. Our approach relies only on basic linear-algebraic properties of similarity transformations of complex matrices. This method recovers the previously known ad hoc results of Dumas, Pernet, and Sedoglavic (2025) and extends them to more general settings, including algorithms involving rational coefficients and square roots, with $i=\sqrt{-1}$ as a special case. Using this framework, we show that no rational scheme is equivalent to Smirnov's $\langle4,4,9,104\rangle$ $\mathbb{Q}[\sqrt{161}]$ algorithm (2022) and that no real scheme is equivalent to the $\langle4,4,4,48\rangle$ complex algorithm of Kaporin (2024). More generally, our approach can also be used to prove the non-existence of integer-coefficient schemes.