Strassen's 2x2 matrix multiplication algorithm: A conceptual perspective

TL;DR

This paper provides a clear, basis-independent, and pedagogical proof of Strassen's 2x2 matrix multiplication algorithm, emphasizing symmetries and algebraic properties to simplify understanding and verification.

Contribution

It introduces a self-contained, coordinate-free proof of Strassen's algorithm, integrating geometric and algebraic perspectives for better pedagogical clarity.

Findings

A basis-independent proof of Strassen's algorithm

Simplification of the verification process

Connection to geometric and algebraic structures

Abstract

The main purpose of this paper is pedagogical. Despite its importance, all proofs of the correctness of Strassen's famous 1969 algorithm to multiply two 2x2 matrices with only seven multiplications involve some basis-dependent calculations such as explicitly multiplying specific 2x2 matrices, expanding expressions to cancel terms with opposing signs, or expanding tensors over the standard basis. This makes the proof nontrivial to memorize and many presentations of the proof avoid showing all the details and leave a significant amount of verifications to the reader. In this note we give a short, self-contained, basis-independent proof of the existence of Strassen's algorithm that avoids these types of calculations. We achieve this by focusing on symmetries and algebraic properties. Our proof can be seen as a coordinate-free version of the construction of Clausen from 1988, combined with recent work on the geometry of Strassen's algorithm by Chiantini, Ikenmeyer, Landsberg, and Ottaviani from 2016.