tensor product


tensor product

[′ten·sər ‚präd·əkt] (mathematics) The product of two tensors is the tensor whose components are obtained by multiplying those of the given tensors. In algebra, a multiplicative operation performed between modules.

tensor product

(mathematics)A function of two vector spaces, U and V,which returns the space of linear maps from V's dual to U.

Tensor product has natural symmetry in interchange of U and Vand it produces an associative "multiplication" on vectorspaces.

Wrinting * for tensor product, we can map UxV to U*V via:(u,v) maps to that linear map which takes any w in V's dual tou times w's action on v. We call this linear map u*v. Onecan then show that

u * v + u * x = u * (v+x)u * v + t * v = (u+t) * vandhu * v = h(u * v) = u * hv

ie, the mapping respects linearity: whence any bilinear map from UxV (to wherever) may be factorised via thismapping. This gives us the degree of natural symmetry inswapping U and V. By rolling it up to multilinear maps fromproducts of several vector spaces, we can get to the naturalassociative "multiplication" on vector spaces.

When all the vector spaces are the same, permutation of thefactors doesn't change the space and so constitutes anautomorphism. These permutation-induced iso-auto-morphismsform a group which is a model of the group ofpermutations.