Ubiquity of the exponent of matrix multiplication

Abstract

The asymptotic exponent of matrix multiplication is the smallest ω such that one may multiply two n × n matrices or invert an n × n matrix in O(nω+e)-complexity for e > 0 arbitrarily small. One of the biggest open problem in complexity theory and numerical linear algebra is its conjectured value ω = 2. This article is about the universality of ω. We will show that ω is not only the asymptotic exponent for the product operation in matrix algebras but also that for various infinite families of Lie algebras, Jordan algebras, and Clifford algebras. In addition, we will show that ω is not just the asymptotic exponent for matrix product and inversion but also that for the evaluation of any matrix-valued polynomial and rational functions of matrix variables.