tensor product

Also found in: Wikipedia.

tensor product

[′ten·sər ‚präd·əkt]
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

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.
References in periodicals archive ?
where [mathematical expression not reproducible] is the symmetric traceless part of the l-fold tensor product of n with itself (see Appendix A, Definition 4 and Section 3).
Among his topics are the categorification of quantum groups, the tensor product of algebras, braiding functors, and rigidity structures.
In this paper, we compute the projective class rings of the tensor product [H.
This process leads to an elegant formulation of 3-D mimetic operators as the tensor product of their 1-D counterparts.
In addition, [cross product] is the structure tensor product.
8-13] estimated error bounds for binary, ternary, quaternary and n-ary curve/surface and tensor product schemes in terms of the maximal differences of the initial control point sequence and constants that depend on the subdivision mask.
Tensor product model transformation in polytopic model-based control.
X](s) for Noetherian schemes via the Kurokawa tensor product of [K1].
In the case of pure states, we say that a given state |[psi]> of n parties is entangled if it is not a tensor product of individual states for each one of the parties, that is,
For two graphs G and H, the tensor product G x H has vertex set V(G) x V(H) in which ([g.
The present paper is a contribution to the growing library of algorithms for the geometric approach, in the case of tensor product groups.
For given non-zero operators T [member of] B(H) and S [member of] B(K), T [cross product] denotes the tensor product on the product space H [cross product] K.