1

I am trying to wrap my head around what exactly a tensor is. I am familiar with the notion of a tensor product, both the basis-dependent and basis-independent definitions, and there seems to be a link between them and tensors. I have seen some sources define tensors as elements of a tensor product, but there seems to be more to it than this. Can a tensor be defined in terms of tensor products, and if so, what else can be said?

I have already gone through the following questions:

but my confusion still remains.

CBBAM
  • 5,883
  • 2
  • 6
  • 18
  • 1
    If $V$ and $W$ are modules, then a tensor is an element of the the tensor product $V\otimes W$. More generally, a tensor is an element of a tensor product of modules. – morrowmh Sep 22 '22 at 22:17
  • @morrowmh That is really all there is to it? – CBBAM Sep 22 '22 at 22:20
  • Note that a simple tensor is an element of $V\otimes W$ of the form $v\otimes w$ where $v\in V, w\in W$. Not all tensors are simple tensors, but they are always linear combinations of simple tensors. See tensor rank for more insights. – hardmath Sep 22 '22 at 22:25
  • @CBBAM In its most abstract form, yes. You have elements of $V$ and elements of $W$, and you want to multiply them. The tensor product $V\otimes W$ gives you a context in which you can do this. – morrowmh Sep 22 '22 at 22:26
  • Have you looked at other posts on this site like this? – blargoner Sep 22 '22 at 22:27
  • 1
    @hardmath So would it be correct to say a tensor is to a tensor product what a vector is to a vector space? – CBBAM Sep 22 '22 at 22:27
  • @blargoner I will take a look at this, thank you. – CBBAM Sep 22 '22 at 22:28
  • https://math.stackexchange.com/questions/3942678/mathematical-properties-of-rank-n-tensors-where-n2/3947326#3947326 – K.defaoite Sep 23 '22 at 10:44

1 Answers1

4

Short answer: To a mathematician, a tensor is just an element of a tensor product. As you ask in the comments, it is correct to say that a tensor is to a tensor product what a vector is to a vector space.

Medium answer: A tremendous amount of confusion here is caused by the fact that the word "tensor" has (at least) five different meanings which are rarely distinguished, and different communities of practitioners (e.g. mathematicians vs. physicists) default to different ones. Here they are:

  1. A tensor is an element of a tensor product.
  2. A tensor of type $(m, n)$ relative to a (usually finite-dimensional) vector space $V$ is an element of the tensor product $V^{\otimes n} \otimes (V^{\ast})^{\otimes m}$.
  3. A tensor is a multidimensional array of numbers (a hypermatrix).
  4. A tensor is a tensor field (e.g. the Riemann curvature tensor or the stress-energy tensor).
  5. A tensor is an object which transforms in a certain way under change of coordinates.

These concepts are of course related (2 is a special case of 1, 3 is how you represent a tensor in sense either 1 or 2 using a choice of basis, 5 is equivalent to either 2 or 4 depending on context, 4 is a generalization of 2) but they are not the same and much confusion results from not carefully distinguishing them.

Additionally sense 2 is more sophisticated than it appears, because we have the freedom to interpret a tensor of type $(m, n)$ as a function between tensor products in $2^{m+n}$ different ways. For example, a tensor of type $(1, 1)$ can be thought of as any of

  • an element of $V \otimes V^{\ast}$,
  • a linear map $V \to V$,
  • a linear map $V^{\ast} \to V^{\ast}$, or
  • a linear functional $V^{\ast} \otimes V \to K$ where $K$ is the underlying field

and any of these representations may be appropriate depending on context.

Qiaochu Yuan
  • 419,620
  • Thanks! I never knew the term is used so ambiguously, that helps explain part of my confusion. – CBBAM Sep 23 '22 at 15:11