# Origin of Tensor Product

When and why did Mathematicians saw a need to define Tensor Products?

I want to know the historical development of the idea "Tensor Product"?

• The prototype for matrices was defined by Zehfuss in 1858, and later mistakenly ascribed to Kronecker by Hensel (it is also called Kronecker product), see On the history of the Kronecker product by Henderson at al.. Zehfuss used it to derive some determinant identities. Nov 15, 2020 at 8:01
• It is posted on ecommons, there are also slides on SIAM, and Schacke gives a historical summary based on it and details on applications in a freely accessible paper. Nov 15, 2020 at 22:22
• For the more abstract modern version of tensor product applied to spaces and groups see Origin of the modern definition of the tensor product on Math SE. It goes back to Whitney (1938) and more generally to Bourbaki (1943). The motivation came from multilinear algebra as applied to homology. Nov 15, 2020 at 23:35
• I think it's more a case of "mathematicians finding another useful shorthand notation for interesting calculations." Analogous to devising exponentiation as shorthand for multiplying Nov 16, 2020 at 13:43

The tensor product is actually a very simple concept.

It goes back to Babylonian times when people realised that two edges describes an area. Intuitively they realised that geometric area was bilinear but all this wasn't formalised until the twentieth century.

Mathematicians, being mathematicians, generalised by allowing the edges to take values in any vector space, as well as the attendant geometric area.

That formalisation took so long shouldn't be such a surprise. After arithmetic was invented well before Babylonian times and yet was only formalised with Peano's axioms in the early 20th C. I'd also add that it has taken the subject so far away from its geometric roots that its difficult to see the geometry of a tensor.

Its worth adding that the tensors of general relativity, which made the subject famous, are actually fields of tensors, that is tensor fields. Moreover, the tensors they use are of a special kind, comprising of a tensor power of a vector space tensored with a tensor power of its dual space.

• @Saikat No, it is not. Bilinearity and tensoring are two different things, and bilinear forms were studied long before tensoring. Tensoring produces only a very special form of bilinearity (rank one). Even speaking loosely, areas and volumes are more related to Grassman's exterior product. Nov 15, 2020 at 8:06
• I’m sorry, but claiming the idea of a tensor product (not just multiplication of numbers, but tensor products of vector spaces) goes back to the Babylonians is unreasonable. Should we say too that the Babylonians were doing calculus because calculus involves area and the area of a very simple region like a rectangle was known to the ancients?
– KCd
Nov 15, 2020 at 8:07
• @Conifold: I don't understand the remark, that tensor products only produce rank one bilinearity. The tensor product $\otimes:V_1\times V_2\to V_1\otimes V_2$ is the universal bilinear map, so it produces all bilinearity in a sense. And Mozibur Ullah is right in that the one dimensional vector space of "areas" is canonically (or by definition?) isomorphic to the tensor product of the one dimensional space of "lengths" with itself. Dec 15, 2020 at 13:39
• That remark might seem trivial if one thinks of lengths and areas as "only numbers" (which requires an arbitrary choice of units of these vector spaces). But as far as a I understand, this identification of geometrical quantities (like lengths and areas) with numbers was not so common in the early history of mathematics. So it is not clear to me, in which sense the observation that the area of a rectangle is bilinear in its square lengths was an obvious triviality for everyone, or a genuine discovery. Dec 15, 2020 at 13:42
• @MoziburUllah you said tensoring produces tensors of rank two and that conifold was wrong to say tensoring produces tensors of rank one. There are two different meanings for the term "rank of a tensor". In a tensor product of $p$ vector spaces, the elements of this space can be called "rank $p$ tensors", e.g., the inner product on $\mathbf R^n$ is a "rank 2 tensor" because it can be viewed in $V \otimes V$ where $V$ is the dual space of $\mathbf R^n$. That is the meaning you have for a tensor. The other meaning for rank of a tensor $t$ is the least number of elementary tensors with sum $t$.
– KCd
Apr 16, 2021 at 17:34