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The border rank of the matrix multiplication operator for n by n matrices is a standard measure of its complexity.
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by same author, The border rank of the multiplication of 2 × 2 2\times 2 matrices is seven , J. Amer. Math. Soc. 19
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by same author, Tensors: Geometry and applications , Graduate Studies in Mathematics, vol. 128, AMS, Providence, 2011
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Virginia V. Williams, Multiplying matrices faster than Coppersmith-Winograd , STOC 2012
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A. Massarenti and E. Raviolo, The rank of n × n n\times n matrix multiplication is at least 3 n 2 − 2 2 n 3 / 2 − 3 n 3n^{2}-2\sqrt{2}n^{3/2}-3n , Linear Algebra and its Applications, 2013, 10.1016/j.laa.2013.01.031, arXiv 1211.6320
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Claudio Procesi, Lie groups , Universitext, Springer, New York, 2007, An approach through invariants and representations. MR MR2265844 (2007j:22016)
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J. Hauenstein, C. Ikenmeyer, and J.M. Landsberg, Computer aided methods for lower bounds on the border rank , arXiv 1305.0779
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J. M. Landsberg, Explicit tensors of border rank at least 2n-1 , preprint arXiv:1209.1664
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J.M. Landsberg, New lower bounds for the rank of matrix multiplication , arXiv 1206.1530
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J.M. Landsberg and Giorgio Ottaviani, Equations for secant varieties of Veronese and other varieties , to appear in Annali di Matematica, doi:10.1007/s10231-011-0238-6, arXiv:1111:4567
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