The tensor product appears as a coproduct for commutative rings with unity, but as with the direct sum this definition is then extended to other categories. For abelian groups, the tensor product \({G\otimes H}\) is the group generated by the ordered pairs \({g\otimes h}\) linear over \({+}\); as more structure is added, the tensor product is required to be bilinear with regard to these structures. It can then be applied to multiple objects by extending these bilinear rules to multilinear ones.
It is helpful to compare the properties of the tensor product to the direct sum in various categories, since consistent with their symbols \({\oplus}\) and \({\otimes}\) they act in many ways like addition and multiplication.
| Direct sum \({\oplus}\) | Tensor product \({\otimes}\) | |
|---|---|---|
| Abelian groups | \({\begin{align*} & (v_{1}\oplus w)+(v_{2}\oplus w)\\ & \equiv(v_{1}+v_{2})\oplus(w+w)\end{align*}}\) | \({\begin{align*} & (v_{1}\otimes w)+(v_{2}\otimes w)\\ & \equiv(v_{1}+v_{2})\otimes w\end{align*}}\) |
| Vector spaces | \({\begin{align*} & a(v\oplus w)\\ & \equiv av\oplus aw\end{align*}}\) | \({\begin{align*} & a(v\otimes w)\\ & \equiv av\otimes w\equiv v\otimes aw\end{align*}}\) |
| Inner product spaces | \({\begin{align*} & \left\langle v_{1}\oplus w_{1},v_{2}\oplus w_{2}\right\rangle \\ & \equiv\left\langle v_{1},v_{2}\right\rangle +\left\langle w_{1},w_{2}\right\rangle \end{align*}}\) | \({\begin{align*} & \left\langle v_{1}\otimes w_{1},v_{2}\otimes w_{2}\right\rangle \\ & \equiv\left\langle v_{1},v_{2}\right\rangle \left\langle w_{1},w_{2}\right\rangle \end{align*}}\) |
| Algebras / Rings | \({\begin{align*} & (v_{1}\oplus w_{1})(v_{2}\oplus w_{2})\\ & \equiv(v_{1}v_{2})\oplus(w_{1}w_{2})\end{align*}}\) | \({\begin{align*} & (v_{1}\otimes w_{1})(v_{2}\otimes w_{2})\\ & \equiv(v_{1}v_{2})\otimes(w_{1}w_{2})\end{align*}}\) |
Notes: The addition and multiplication of inner products is that of scalars, while the multiplication of vectors is that of the algebra or ring.
It is important to remember that elements of the direct sum \({V\oplus W}\) always have the form \({v\oplus w}\), while elements of the tensor product \({V\otimes W}\) are generated by the elements \({v\otimes w}\) using the operation \({+}\) as defined above, so that the general element of \({V\otimes W}\) has the form of a sum \({\sum\left(v_{\mu}\otimes w_{\nu}\right)}\). For example, if \({V}\) and \({W}\) are \({m}\)- and \({n}\)-dimensional vector spaces with bases \({d_{\mu}}\) and \({e_{\nu}}\), \({V\otimes W}\) has basis \({\left\{ d_{\mu}\otimes e_{\nu}\right\} }\) and dimension \({mn}\), while \({V\oplus W}\) has basis \({\left\{ d_{1},\dotsc,d_{m},e_{1},\dotsc,e_{n}\right\}}\) and dimension \({m+n}\). If \({V}\) and \({W}\) are algebras defined by square matrices, the direct sum \({V\oplus W}\) and tensor product \({V\otimes W}\) have elements that are isomorphic to matrices that can be formed from the matrices \({v}\) and \({w}\):
\(\displaystyle v\oplus w\cong\begin{pmatrix}v & 0\\ 0 & w \end{pmatrix}\quad v\otimes w\cong\begin{pmatrix}v_{11}w & v_{12}w & \cdots\\ v_{21}w & v_{22}w & \cdots\\ \vdots & \vdots & \ddots \end{pmatrix} \)
The convention used in the second isomorphism, in which \({v\otimes w}\) is “the matrix \({v}\) with elements multiples of \({w}\),” is sometimes called the Kronecker product; one can also choose to use the opposite convention. Some specific isomorphisms (as real algebras) include:
- \({\mathbb{C}\otimes\mathbb{C}\cong\mathbb{C}\oplus\mathbb{C}}\)
- \({\mathbb{C}\otimes\mathbb{H}\cong\mathbb{C}\left(2\right)}\)
- \({\mathbb{H}\otimes\mathbb{H}\cong\mathbb{R}\left(4\right)}\)
where e.g. \({\mathbb{C}\left(2\right)}\) denotes the algebra of complex \({2\times2}\) matrices. Complexification is equivalent to tensoring with the complex numbers, i.e. \({V^{\mathbb{C}}\cong V\otimes\mathbb{C}}\), so the first isomorphism can be viewed as the complexification of \({\mathbb{C}}\) as a real algebra. An explicit isomorphism is \({a(1\otimes1)+b(i\otimes1)+c(1\otimes i)+d(i\otimes i)\mapsto\left((a+d)+i(b-c),(a-d)+i(b+c)\right)}\), or in the reverse direction \({(z,w)\mapsto\frac{z}{2}\left(1\otimes1+i\otimes i\right)+\frac{w}{2}\left(1\otimes1-i\otimes i\right)}\). Note that the original algebra is thus embedded as \({a+ib\mapsto(a+ib,a+ib)}\). We can then apply this isomorphism to each matrix element in \({\mathbb{C}(n)}\) as a real algebra to get \({\mathbb{C}(n)^{\mathbb{C}}\cong\mathbb{C}(n)\otimes\mathbb{C}\cong\mathbb{C}(n)\oplus\mathbb{C}(n)}\), where again uncomplexified elements are mapped as \({v\mapsto(v,v)}\).