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Strassen’s 2 × 2 matrix multiplication algorithm: a conceptual perspective

Publikation: Bidrag til tidsskriftTidsskriftartikelForskningfagfællebedømt

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, sometimes involving clever simplifications using the sparsity of tensor summands. 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.

OriginalsprogEngelsk
TidsskriftAnnali dell'Universita di Ferrara
Vol/bind65
Udgave nummer2
Sider (fra-til)241-248
Antal sider8
ISSN0430-3202
DOI
StatusUdgivet - 1 nov. 2019
Eksternt udgivetJa

ID: 232711356