TY - JOUR

T1 - Strassen’s 2 × 2 matrix multiplication algorithm

T2 - a conceptual perspective

AU - Ikenmeyer, Christian

AU - Lysikov, Vladimir

PY - 2019/11/1

Y1 - 2019/11/1

N2 - 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.

AB - 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.

KW - Coordinate-free

KW - Elementary

KW - Matrix multiplication

KW - Strassen’s algorithm

UR - http://www.scopus.com/inward/record.url?scp=85067808056&partnerID=8YFLogxK

UR - https://arxiv.org/abs/1708.08083

U2 - 10.1007/s11565-019-00318-1

DO - 10.1007/s11565-019-00318-1

M3 - Journal article

AN - SCOPUS:85067808056

VL - 65

SP - 241

EP - 248

JO - Annali dell'Universita di Ferrara

JF - Annali dell'Universita di Ferrara

SN - 0430-3202

IS - 2

ER -