Ranks derived from multilinear maps
Let V1, V2 and V3 be vector spaces over any field k. An element Tset membership, variantV1circle times operatorV2circle times operatorV3 induces for each i=1,2,3 a k-linear map View the MathML source where View the MathML source is the dual vector space of Vi. We characterize all integer triplets (r1,r2,r3) such that there exists a tensor T with ri=rankTi, and we explain how these ranks are related to the higher secant varieties of various Segre varieties. We also study the case Tset membership, variantV1circle times operatorcdots, three dots, centeredcircle times operatorVn with n>3, giving necessary conditions on the ranks of all induced linear maps.
Publisert i Journal of Pure and Applied Algebra, 2011
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