Complex to Rational Fast Matrix Multiplication

2026-02-13

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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.

Problem

• fast matrix multiplication
• complex coefficients
• rational algorithms
• real coefficients
• matrix multiplication schemes

Innovation

• fast matrix multiplication
• rational coefficients
• complex-to-real transformation
• similarity transformation
• non-existence proof