# I Rank-2 Tensor

1. May 12, 2017

### 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?

2. May 12, 2017

### Staff: Mentor

Yes. This assumes, however, the choice of some basis, according to which the matrix entries are the coordinates.
It's the same as between $f : U^* \times V^* \rightarrow \mathbb{F}\, , \,f : U \times V \rightarrow \mathbb{F}\, , \,f: U^* \times V \rightarrow \mathbb{F}$.
Yes.
requires a basis of both ...
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?
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 $(1,2)$.
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.

3. May 12, 2017

### pervect

Staff Emeritus
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.

4. Jun 22, 2017