Talk:Tensor

Change Algebraic Object to Algebraic Structure in Tensor Definition

I propose changing the wording in the definition of tensor from using the term algebraic object to algebraic structure or to change the link destination to the entry for algebraic structure. As of now the 'algebraic object' links to the mathematical object entry, which is not relevant to the the definition of a tensor in a direct way. Linking to the algebraic structure entry and perhaps changing the text itself would have more utility and makes more sense. 69.162.230.42 (talk) 21:47, 14 December 2023 (UTC)

I would reserve "structure" for the system into which a tensor fits: $(V^{*})^{\otimes q}\otimes V^{\otimes p}$, for a (p, q)-tensor. So I disagree, but I don't feel strongly about it. Mgnbar (talk) 23:16, 14 December 2023 (UTC)

An algebraic structure is a set equipped with some operations. Not being a set, a tensor cannot be an algebraic structure. So the lead is correct by saying that a tensor is an algebraic object. Indeed, tensors are mathematical objects that belong to some algebraic structures (tensor algebra or geometric algebra). So, I strongly disagree with your suggestion that goes against the standard mathematical terminology. D.Lazard (talk) 09:51, 15 December 2023 (UTC)

Seeming definition unclarity

In the section "As multilinear maps", a (p,q) tensor is defined w/r/t the (direct) product of p copies of V* and q copies of V, but in the section "Using tensor products", a (p,q) tensor is defined as the tensor product of p copies of V and q copies of V*. Can an explanation about why p and q flip roles be added to one of these sections? Makhnoboi19 (talk) 17:18, 7 January 2024 (UTC)

The reason is that, in the "As multilinear maps" section, we are considering tensors as maps from the vector space in question to R. That is, a tensor is not an element of the vector space in question, but rather of its dual. And this dual-ness shows up as the "opposite" vector space in the section "Using tensor products".

I agree that this might deserve clarification in the article. But I'm not sure what to write. Does anyone want to take a stab at it? Mgnbar (talk) 22:33, 7 January 2024 (UTC)

The bulk of this article should not use the Einstein summation convention

The bulk of this article should not use the Einstein summation convention. The Einstein summation convention should be introduced in a special section near the bottom of the article. 134.16.68.202 (talk) 01:22, 25 January 2024 (UTC)

I tend to agree, although I don't feel strongly. There are a few cases (general transformation rule) where explicitly showing the summations would significantly expand the length of the expression. But it would be a net benefit to novice readers. Mgnbar (talk) 21:07, 25 January 2024 (UTC)

Incorrect definition

The definitions in the "as multilinear maps" and "using tensor products" sections are wrong. Requiring all copies of to be the same vector space would exclude non-square matrices from being tensors (a $2\times 3$ matrix for example is a multilinear map $\mathbb {R} ^{2}\times \mathbb {R} ^{3}\to \mathbb {R}$). 65.93.245.82 (talk) 06:09, 23 April 2024 (UTC)

The definition is not so much incorrect as restrictive. This article views tensors in terms of a single vector space V and its dual V*. In the past, we have had discussions about whether "tensor" simply means "element of a tensor product of vector spaces", which would be broad enough to handle your objection. The consensus was that the term "tensor" is not used that broadly in the literature. Mgnbar (talk) 11:46, 23 April 2024 (UTC)