A (5/2)n2-Lower Bound for the Multiplicative Complexity of n×n-Matrix Multiplication

Authors:
Markus Bläser
Affiliations:
-
Venue:
STACS '01 Proceedings of the 18th Annual Symposium on Theoretical Aspects of Computer Science
Year:
2001

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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.