A 5/2n2-lower bound for the multiplicative complexity of n × n-matrix multiplication

Abstract

We prove a lower bound of 5/2n2 − 3n for the multiplicative complexity of n × n-matrix multiplication over arbitrary fields. More general, we show that for any finite dimensional semisimple algebra A with unity, the multiplicative complexity of the multiplication in A is bounded from below by 5/2 dim A − 3(n1 + … + nt) if the decomposition of A ≌ A1 ×… × At into simple algebras Aτ ≌ Dτnτ×nτ contains only noncommutative factors, that is, the division algebra Dτ is noncommutative or nτ ≥ 2.