The Geometry of Rank Decompositions of Matrix Multiplication I: 2 × 2 Matrices

ABSTRACT

This is the first in a series of papers on rank decompositions of the matrix multiplication tensor. In this paper, we establish general facts about rank decompositions of tensors, describe potential ways to search for new matrix multiplication decompositions, give a geometric proof of the theorem of Burichenko establishing the symmetry group of Strassen's algorithm, and present two particularly nice subfamilies in the Strassen family of decompositions.

Keywords:

• Matrix multiplication
• tensor rank
• tensor decomposition
• symmetries

2000 AMS SUBJECT CLASSIFICATION:

• 68Q17
• 14L30
• 15A69

Acknowledgments

This project began at a November 2013 “Research in pairs” program at Mathematisches Forschungsinstitut Oberwolfach. The authors thank the institute for its hospitality and a great work environment. Christian Ikenmeyer was at Texas A&M University during most of the time this research was conducted.

Additional information

Funding

Division of Mathematical Sciences (1405348)

J. M. Landsberg is supported by NSF grant DMS-1405348.