These lecture notes are intended as an introduction to several notions of tensor rank and their connections to the asymptotic complexity of matrix multiplication. The latter is studied with the exponent of matrix multiplication, which will be expressed in terms of tensor (border) rank, (border) symmetric rank and the asymptotic rank of certain tensors. We introduce the multilinear rank of a tensor as well, deal with the concept of tensor equivalence and study prehomogeneous vector spaces with the Castling transform. Moreover, we treat Apolarity Theory and use it to determine the symmetric rank (Waring rank) of some symmetric tensors.

中文翻译：

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张量等级和复杂度

这些讲义旨在介绍张量秩的几个概念及其与矩阵乘法渐近复杂性的联系。后者是用矩阵乘法的指数来研究的，它将用张量（边界）秩、（边界）对称秩和某些张量的渐近秩来表示。我们还介绍了张量的多线性秩，处理了张量等价的概念，并使用 Castling 变换研究了前齐次向量空间。此外，我们处理 Apolarity Theory 并用它来确定一些对称张量的对称秩（Waring rank）。