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

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, ex...

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Vydáno v:Annali dell'Università di Ferrara. Sezione 7. Scienze matematiche Ročník 65; číslo 2; s. 241 - 248
Hlavní autoři: Ikenmeyer, Christian, Lysikov, Vladimir
Médium: Journal Article
Jazyk:angličtina
Vydáno: Milan Springer Milan 01.11.2019
Springer Nature B.V
Témata:
Algebraic Geometry
Algorithms
Analysis
Geometry
History of Mathematical Sciences
Mathematical analysis
Mathematics
Mathematics and Statistics
Matrices (mathematics)
Multiplication
Numerical Analysis
Tensors
Coordinate-free
Elementary
Matrix multiplication
68W30 Symbolic computation and algebraic computation
Strassen’s algorithm
ISSN:0430-3202, 1827-1510
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Shrnutí: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.
Bibliografie:ObjectType-Article-1
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ISSN:0430-3202
1827-1510
DOI:10.1007/s11565-019-00318-1