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

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 2 × 2 matrices with only seven multiplications involve some basis-dependent calculations such as explicitly multiplying specific 2×2 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.

Designing Strassen's algorithm

“…However, that construction relies on a seemingly magical property of a certain 4 × 4 multiplication table. More recently, Ikenmeyer and Lysikov [IL17] gave a beautiful explanation of Clausen's construction, but ultimately their proof for Strassen's algorithm still relies on the same magical property of the same 4 × 4 multiplication table, and it is not immediately obvious how to generalize to all n. In contrast, our result easily generalizes to all n, and more generally to orbits of any irreducible representation of any finite group.…”