# tensor product

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## 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.
This article is provided by FOLDOC - Free Online Dictionary of Computing (foldoc.org)
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In algebra it is known that, for any finite collection [V.sub.1],...,[V.sub.n] of linear spaces, their tensor product [V.sub.1] [cross product] xxx [cross product] [V.sub.n] is a linear space and consists of all finite sums of the form
Then we construct the multi-dimensional B-spline interpolation operator [mathematical expression not reproducible] by the tensor product
Recently, Shao  defined the general product of two n-dimensional tensors as follows, and one of the applications of the tensor product is that [Ax.sup.m-1] can be simply written as A x x.
Khoromskij, "Simultaneous state-time approximation of the chemical master equation using tensor product formats," Numerical Linear Algebra with Applications, vol.
Moreover, the properties of the mode tensor product are provided below:
The notions of normal product and tensor product of m-polar fuzzy graphs are introduced and some properties are studied.
Over the years, researchers have focused on the implementation of Chebyshev tensor product series in image analysis [5-7].
It is shown in [19, 28] that the Drinfeld double D(H) is a cocycle twisting of the tensor product Hopf algebra [H.sup.[*cop]] [cross product] H.
The graph G is the tensor product G x [K.sub.2] of G with the connected graph [K.sub.2] with 2 vertices; the vertex set of G x [K.sub.2] is the Cartesian product of the vertices of G and [K.sub.2], there are edges in G x [K.sub.2] between (a, 0) and (b,1) and between (a, 1) and (b,0) if and only if there is an edge in G between a and b; see [HIK11].
In addition, [cross product] is the structure tensor product. The image gradients [I.sub.x] ([P.sub.[sigma]],) and [I.sub.y] ([P.sub.[sigma]'],) can be used in x and y directions.
[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.
That is, a tensor product between an arbitrary rank n tensor and [[LAMBDA].sup.(n)] will extract the symmetric part of that tensor.

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