# tensor product

Also found in: Wikipedia.

## 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.

McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.

## 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 V and it produces an associative "multiplication" on vector spaces.

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 to u times w's action on v. We call this linear map u*v. One can then show that

u * v + u * x = u * (v+x) u * v + t * v = (u+t) * v and hu * v = h(u * v) = u * hv

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

When all the vector spaces are the same, permutation of the factors doesn't change the space and so constitutes an automorphism. These permutation-induced iso-auto-morphisms form a group which is a model of the group of permutations.

Tensor product has natural symmetry in interchange of U and V and it produces an associative "multiplication" on vector spaces.

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 to u times w's action on v. We call this linear map u*v. One can then show that

u * v + u * x = u * (v+x) u * v + t * v = (u+t) * v and hu * v = h(u * v) = u * hv

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

When all the vector spaces are the same, permutation of the factors doesn't change the space and so constitutes an automorphism. These permutation-induced iso-auto-morphisms form a group which is a model of the group of permutations.

This article is provided by FOLDOC - Free Online Dictionary of Computing (

**foldoc.org**)Want to thank TFD for its existence? Tell a friend about us, add a link to this page, or visit the webmaster's page for free fun content.

Link to this page: