Getting to Grips with Rank-2 Tensors

[Silviu]

Hello! I am reading about tensors and I am a bit confused about rank-2 tensors. From what I understand they can be represented by a matrix. However I am not sure I understand the difference between (2,0), (0,2) and (1,1) tensors. I understand that they act on different objects (vectors or one forms or both) but having a matrix, how can u know what kind of tensor it is? Also, for example I saw definitions of electromagnetic field strength tensor written both as (2,0) and (0,2) tensor and I am not sure I understand the difference. What would be the 2 vectors it can act on and what would be the 2 one-forms it can act on, and how do we know when to use the right form?

 

[fresh_42]

Hello! I am reading about tensors and I am a bit confused about rank-2 tensors. From what I understand they can be represented by a matrix.

Yes. This assumes, however, the choice of some basis, according to which the matrix entries are the coordinates.

However I am not sure I understand the difference between (2,0), (0,2) and (1,1) tensors.

It's the same as between $f : U^* \times V^* \rightarrow \mathbb{F}\, , \,f : U \times V \rightarrow \mathbb{F}\, , \,f: U^* \times V \rightarrow \mathbb{F}$.

I understand that they act on different objects (vectors or one forms or both) ...

Yes.

... but having a matrix, ...

requires a basis of both ...

how can u know what kind of tensor it is?

You can't. How can you tell, whether $(1,2)$ is a vector, a linear function $f : \mathbb{R}^2 \rightarrow \mathbb{R}\, , \,x \mapsto \langle (1,2),x\rangle$ or simply a point in the Euclidean plane? Or a lattice point?

Also, for example I saw definitions of electromagnetic field strength tensor written both as (2,0) and (0,2) tensor and I am not sure I understand the difference.

As $V^* \cong V$ the difference is, whether $(1,2) \in V$ or $(x \mapsto \langle (1,2),x \rangle) \in V^*$. Both are represented by $(1,2)$.

What would be the 2 vectors it can act on and what would be the 2 one-forms it can act on, and how do we know when to use the right form?

It simply depends on what you want the matrix $M$ to represent. You have $f(u,v)= u^tMv$ and from which vector spaces $u$ and $v$ are, depends on what you want to do.

 

[pervect]

Hello! I am reading about tensors and I am a bit confused about rank-2 tensors. From what I understand they can be represented by a matrix. However I am not sure I understand the difference between (2,0), (0,2) and (1,1) tensors. I understand that they act on different objects (vectors or one forms or both) but having a matrix, how can u know what kind of tensor it is? Also, for example I saw definitions of electromagnetic field strength tensor written both as (2,0) and (0,2) tensor and I am not sure I understand the difference. What would be the 2 vectors it can act on and what would be the 2 one-forms it can act on, and how do we know when to use the right form?

Matrix notion is simply not as powerful as tensor notation. It may be helpful, though, to regard tensor vectors as matrix column vectors, and tensor one-forms as matrix row vectors. Then the the product of a row and column vector yields a scalar, which is what a vector and a one form written in tensor notation do.

Then the typical matrix is a linear map from a column vector to a column vector. You can also regard it as a map from a row vector to a row vector, though this is less common.

Maps from vectors to one forms, and one-forms to vectors exist in tensor notation (the metric tensor is one example of this). But it doesn't really have a direct ananlogy in matrix form, though the metric tensor $g_{\mu\nu}$ is sometimes written to appear as a matrix.

 

[scottdave]

This thread, along with the article (I think it is one of the Insights) has stirred my interest to learn more about tensors (I already knew a little). In my searches, I came across the following blog articles, which I found interesting.
https://www.naturelovesmath.com/en/mathematics/tensors-for-the-noob/
and another by the same person with more detail:
https://www.naturelovesmath.com/en/mathematics/about-numbers-episode-5-tensors/