Talk:Tensor contraction

Abstract Indices

I think it's better to avoid using four-dimensional (or any specific dimension) tensors since this is often against the modern use of tensors (the abstract index concept of tensors). --Freiddie 22:00, 10 April 2007 (UTC)

Major rewrite

In the spirit of "be bold", I did a lot of work on this article, moving stuff around and rewriting a lot. In particular, I took out all of the 0...3 stuff that is specific to physics, and I put the abstract definition first. I also removed a lot of redundancy; e.g. the (1,1) case was discussed in about 4 different ways previously. Please respond here if you have issues with the rewrite (which is still a work in progress). Thanks -- Spireguy (talk) 19:43, 6 March 2008 (UTC)

Abstract formulation

I removed the following section, because as far as I can see this definition is wrong.

Let V be a vector space over a field k. The core of the contraction operation, and the simplest case, is the natural pairing of V with its dual vector space V*. The pairing is the linear transformation

$C:V^{*}\otimes V\rightarrow k$

corresponding to the bilinear form

$\langle f,v\rangle =f(v)$

where f is in V* and v is in V. The map C defines the contraction operation on a tensor of type (1,1), which is an element of $V^{*}\otimes V$. Note that the result is a scalar (an element of k). Using the natural isomorphism between $V^{*}\otimes V$ and the space of linear transformations from V to V, one obtains a basis-invariant definition of the trace.

In general, a tensor of type (m, n) (with m ≥ 1 and n ≥ 1) is an element of the vector space

$V\otimes \dots \otimes V\otimes V^{*}\otimes \dots \otimes V^{*}$

(where there are m V factors and n V^* factors). Applying the natural pairing to the kth V factor and the lth V^* factor, and using the identity on all other factors, defines the (k, l) contraction operation, which is a linear map which yields a tensor of type (m-1, n-1). By analogy with the (1,1) case, the general contraction operation is sometimes called the trace.

The tensor of type (m, n) is not an element of $V\otimes \dots \otimes V\otimes V^{*}\otimes \dots \otimes V^{*}$, but is the linear form over this space. In particular there is no isomorphism between $V^{*}\otimes V$ and the space of linear transformations from V to V. (The first space has dimension 2n, and the second is n^2).

Please confirm that, or correct me, if I'm wrong.

Is there a correct abstract definition of the contraction? Alexei Kopylov (talk) 21:40, 24 July 2008 (UTC)

Sorry, but the abstract formulation given is completely correct and standard. I didn't specifically put in a cite for that section, since it is so standard, but I can put one in if desired, e.g. Warner's book on manifolds. Let me try to address what I think are the two main confusions.

(1) In the abstract treatment of tensors (see e.g. Component-free treatment of tensors), tensors need not be defined as linear maps. They are simply elements of some tensor product space, e.g. the m+n-fold tensor product given above. E.g. if V is the tangent space to a manifold, then a tensor of type (1,0) (purely contravariant) is a vector itself, i.e. an element of V, and a tensor of type (0,1) is dual vector (1-form), i.e. an element of V^*. Tensors can be described as linear maps, if desired, but that is not necessary: e.g. a (1,0)-tensor (an element of V itself) can be described as a linear functional on V^* (by the double-dual isomorphism), which is a very useful fact, but not the simplest description of an element of V.

(2) The claim that there is no isomorphism between $V^{*}\otimes V$ and the space of linear transformations from V to V is incorrect. First of all, the dimension count given is incorrect; the dimension is multiplicative under tensor product. The natural isomorphism $V^{*}\otimes V$ --> L(V,V) is given by (f,v) --> g, where g(w) = f(w)v. (The product in the last expression is scalar multiplication.) This is quite standard.

So, I'm going to reinstate the section as it was, but the fact that Alexei found it confusing might mean that it needs to be clarified or expanded, or at least that there should be wikilinks to facts such as the explicit definition of the natural isomorphism given just above. I'm also open to a discussion of whether it should be placed before the index treatment. I put it there since I'm a mathematician, and I like having the most elegant definition first, even if some readers will have to skim it and dig into a later section for a more down-to-earth treatment. But I could see switching the order. Comments? -- Spireguy (talk) 02:36, 25 July 2008 (UTC)

Sorry, I was wrong. I confused tensor product and direct sum. :(

I think it would help to add links to Component-free treatment of tensors and Tensor product#Tensor product of vector spaces or Component-free treatment of tensors#Definition: Tensor Product of Vector Spaces.

I don't think switching the order is necessary. I'd also like to see the abstract definition first. Thanks, Alexei Kopylov (talk) 06:35, 25 July 2008 (UTC)

Colon notation

It would be good to have some mention of the colon notation for contractions, commonly employed in mechanics. $A\cdot B$ indicates contraction over one index, $A:B$ is a contraction over two indices, and $A::B$ is a contraction over four indices. (For a random example, see [1].) — Steven G. Johnson (talk) 20:11, 24 August 2009 (UTC)

Most Simplistic Examples Needed

I understand that Wikipedia is more of an encyclopedic formulation of things. However, I note that often articles are excessively abstract and general without giving simplistic strait forward concrete examples that do exist. Admittedly this article is maybe better than average on being readable. That said I would like to see some easy examples of tensor contraction. They might not be the most general formalization and certainly not the most abstract description, but they would be useful to those unfamiliar with the subject. After all tensors were not developed in the abstract. They began first with scalars then with vectors then with matrices and so on. Why not show tensor contraction with a matrix and then with a system of rank 3? — Preceding unsigned comment added by Berrtus (talk • contribs) 00:43, 28 April 2013 (UTC)

Historical Introduction

I'm also a mathematician, if a bit out of date, but I think articles like this need a bit of historical context to motivate the abstract definitions. For example, what problem was index contraction originally developed to solve? UrbanCyborg (talk) 16:26, 2 June 2019 (UTC)

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Bold 37.111.137.166 (talk) 06:27, 17 March 2024 (UTC)