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

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STACS 2001 (STACS 2001)

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Abstract

We prove a lower bound of 5/2n 2-3n 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(n 1 + ⋯ + n t) if the decomposition of A ≅= A1 x ... x At into simple algebras Aτ ≅ Dτnτ×nτ contains only noncommutative factors, that is, the division algebra Dτ is noncommutative or nτ ≥ 2.

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Author information

Authors and Affiliations

  1. Institut für Theoretische Informatik, Med. Universität zu Lübeck, Wallstr. 40, 23560, Lübeck, Germany

    Markus Bläser

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  1. Markus Bläser
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Editors and Affiliations

  1. I3S & INRIA Sophia Antipolis INRIA, CNRS, 2004 Route des Lucioles, 06902, Sophia-Antipolis, France

    Afonso Ferreira

  2. Institut für Theoretische Informatik, Fakultät Informatik, TU Dresden, 01062, Dresden, Germany

    Horst Reichel

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Bläser, M. (2001). A 5/2n 2-Lower Bound for the Multiplicative Complexity of n × n-Matrix Multiplication. In: Ferreira, A., Reichel, H. (eds) STACS 2001. STACS 2001. Lecture Notes in Computer Science, vol 2010. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44693-1_9

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  • DOI: https://doi.org/10.1007/3-540-44693-1_9

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-41695-1

  • Online ISBN: 978-3-540-44693-4

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