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

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

Saikat
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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.

Mozibur Ullah
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  • So is Tensor Product generalization of Geometric Area? – Saikat Nov 15 '20 at 06:54
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    @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. – Conifold Nov 15 '20 at 08:06
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    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 '20 at 08:07
  • Actually I wanted to know the historical development of the idea "Tensor Product", not in terms of the answer provided above. But in terms of modern mathematics. I mean to say what concept and idea in pure math gave rise to idea of Tensor Products? – Saikat Nov 15 '20 at 09:14
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    @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. – Michael Bächtold Dec 15 '20 at 13:39
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    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. – Michael Bächtold Dec 15 '20 at 13:42
  • @Michael Bachtold: The area of a rectangle, the simplest possible geometry of which we want the area of and which we model areas by is ab, where a and b are the lengths of the sides. This is obviously bilinear. – Mozibur Ullah Dec 15 '20 at 15:31
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    It's obviously bilinear from our perspective (as it's trivial for us to acvept 0 or the negative numbers etc ) but first you have to come up with the idea of defining (or identifying) area with ab, and I was wondering if this was a trivial step in the development of geometry/mathematics. – Michael Bächtold Dec 15 '20 at 15:38
  • @Michael Bachtold: I think the origin of measuring rectangular area is lost to time. It most likely predates the Babylonians. Its not a trivial step, but nor then was counting. Its trivial for us, but thats being anachronistic. – Mozibur Ullah Dec 15 '20 at 16:46
  • @KCd: Of a primitive sort, yes. Everything has antecedents. After all, Archimedes is said to have used limiting arguments in calculating areas and hence said to have a notion of calculus. The bilinearity of area is obvious and its what defines the tensor product. This geometric way of thinking about the tensor product is not emphasised enough in my opinion. – Mozibur Ullah Apr 16 '21 at 17:14
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    @conifold: No, you're wrong. Tensoring two vectors produces tensors of rank two. Its a vector that is a tensor of rank one. And its easy to see that tensoring three edges gives volumes. The ancients would have stopped there because their geometric imagination would have been limited by physical 3d dimensions. The modern concept of tensors frees this dimensionality restriction. Nevertheless, the Babylonians understood the concept of tensor, even though they would mot have termed it like this. In my opinion, this geometric understanding of tensors is much under appreciated ... – Mozibur Ullah Apr 16 '21 at 17:19
  • @conifold: Generally speaking, the modern pedagogy of tensors leaves much to be desired. – Mozibur Ullah Apr 16 '21 at 17:20
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    @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 '21 at 17:34
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    In this second sense, a nonzero elementary tensor in any tensor product of vector spaces has rank 1. I think that is what conifold meant by "rank": tensoring directly produces elementary tensors, hence tensors of rank 1. When you view a linear map $f \colon V \to W$ as a tensor in $V^* \otimes W$, its rank in the second sense is the dimension of $f(V)$, which is how the term "rank" is used in linear algebra (dimension of the image). Computing tensor rank in this second sense is a famous hard problem once the number of spaces in the tensor product is bigger than two. – KCd Apr 16 '21 at 17:35
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    @KCd: Looking at Conifolds comment, I think you are right to say he's using the second definition. I have also come across this notion of rank. However, it is the first definition of rank that is used in textbooks on General Relativity and the subject which made the topic of tensors famous. Hence, I think this is the one to think about first and why I did so. I'd also add that Conifold is wrong to think that bilinearity and tensoring are completely different. Tensors are defined in terms of bilinearity in Bourbaki. – Mozibur Ullah Apr 16 '21 at 17:43
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    @conifold: See above. – Mozibur Ullah Apr 16 '21 at 17:43