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.

翻译：这些讲演说明旨在介绍若干高压等级概念及其与矩阵倍增的无症状复杂性的联系,后者与矩阵倍增的引言一起研究,矩阵倍增将以某些高压(边境)等级、(边境)对称等级和某些高压的无症状等级表示,我们还引入了高压的多线等级,处理高等概念,研究高等等概念,并研究与挤塑变异的超异矢量空间。此外,我们处理极化理论,并用它来确定一些对称的温度等级(作战等级)。