Tensor Product

Abstract

In mathematics, the tensor product V ⊗ W of two vector spaces V and W (over the same field) is itself a vector space, endowed with the operation of bilinear composition, denoted by ⊗, from ordered pairs in the Cartesian product V × W onto V ⊗ W in a way that generalizes the outer product. Essentially the difference between a tensor product of two vectors and an ordered pair of vectors is that if one vector is multiplied by a nonzero scalar and the other is multiplied by the reciprocal of that scalar, the result is a different ordered pair of vectors, but the same tensor product of two vectors. The tensor product of V and W is the vector space generated by the symbols v ⊗ w, with v ∈ V and w ∈ W, in which the relations of bilinearity are imposed for the product operation ⊗, and no other relations are assumed to hold. The tensor product space is thus the "freest" (or most general) such vector space, in the sense of having the fewest constraints. The tensor product of (finite-dimensional) vector spaces has dimension equal to the product of the dimensions of the two factors: dim ⁡ ( V ⊗ W ) = dim ⁡ V × dim ⁡ W . {\displaystyle \dim(V\otimes W)=\dim V\times \dim W.} In particular, this distinguishes the tensor product from the direct sum vector space, whose dimension is the sum of the dimensions of the two summands: dim ⁡ ( V ⊕ W ) = dim ⁡ V + dim ⁡ W . {\displaystyle \dim(V\oplus W)=\dim V+\dim W.} More generally, the tensor product can be extended to other categories of mathematical objects in addition to vector spaces, such as to matrices, tensors, algebras, topological vector spaces, and modules. In each such case the tensor product is characterized by a similar universal property: it is the freest bilinear operation. The general concept of a "tensor product" is captured by monoidal categories; that is, the class of all things that have a tensor product is a monoidal category.