On the rank of $n\times n$ matrix multiplication

Abstract

For every $p\le n$ positive integer we obtain the lower bound $(3-\frac{1}{p+1})n^2-\left(2\right(\genfrac{}{}{0ex}{}{2p}{p+1})-(\genfrac{}{}{0ex}{}{2p-2}{p-1})+2)n$ for the rank of the $n\times n$ matrix multiplication. This bound improves the previous one $(3-\frac{1}{p+1})n^2-(1+2p(\genfrac{}{}{0ex}{}{2p}{p}\left)\right)n$ due to Landsberg. Furthermore our bound improves the classic bound $\frac{5}{2}{n}^2-3n$, due to Bläser, for every $n\ge 132$. Finally, for $p=2$, with a sligtly different strategy we menage to obtain the lower bound $\frac{8}{3}{n}^2-7n$ which improves Bläser's bound for any $n\ge 24$.

10 pages. New version, title and main result changed. Linear Algebra and its Applications 2013

References in corpus (1)

• New lower bounds for the border rank of matrix multiplication

Cited by in corpus (1)

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