Solution: What is Decomposable Tensor?

[Sudharaka]

Hi everyone, :)

Here's a question I am trying to solve at the moment. I want to know what is meant by decomposable in this context. Really appreciate any input. :)

Problem:

Let \(V\) be a vector space over a field \(F\), \(\{e_1,\,e_2,\,\cdots,\,e_n\}\) a basis in \(V\) and \(\{e^1,\,\cdots,\,e^n\}\) adjoint basis in \(V^*\). Which of the following tensors, given by their coordinates are decomposable:

a) \[t^k_{ij}=2^{i+j+k^2}\]

b) \[t^{ij}=i+j\]

 

[Opalg]

I don't know for sure, but I'm guessing that a decomposable tensor would be what I would call a rank-one tensor, in other words one of the form $x\otimes y$ for some $x\in V$, $y\in V^*$. (One of those two given tensors has rank 1, the other one does not.)

 

[Sudharaka]

I don't know for sure, but I'm guessing that a decomposable tensor would be what I would call a rank-one tensor, in other words one of the form $x\otimes y$ for some $x\in V$, $y\in V^*$. (One of those two given tensors has rank 1, the other one does not.)

Thanks for the reply, but isn't \(t^{k}_{ij}\) a rank three tensor and \(t^{ij}\) a rank two tensor. Sorry, but I am new to tensors and I always view the rank as the number of indices. :)

 

[Opalg]

Thanks for the reply, but isn't \(t^{k}_{ij}\) a rank three tensor and \(t^{ij}\) a rank two tensor. Sorry, but I am new to tensors and I always view the rank as the number of indices. :)

One of the difficulties with tensor products is that they crop up in very different contexts and people use different notations and terminology. I think of a tensor $t^{ij}$ as being represented by a matrix. When I call it a rank-one tensor I am thinking of the rank of the associated matrix, which is obviously different from your usage of the word "rank" here.

The reason that I instinctively feel that your tensor $t_{ij}^k = 2^{i+j+k^2}$ is decomposable is that it "decomposes" as a product $t_{ij}^k = 2^{i}\times2^{j+k^2}$. But since I am unsure about the notation you are using, I won't attempt to say more.

 

[Sudharaka]

One of the difficulties with tensor products is that they crop up in very different contexts and people use different notations and terminology. I think of a tensor $t^{ij}$ as being represented by a matrix. When I call it a rank-one tensor I am thinking of the rank of the associated matrix, which is obviously different from your usage of the word "rank" here.

The reason that I instinctively feel that your tensor $t_{ij}^k = 2^{i+j+k^2}$ is decomposable is that it "decomposes" as a product $t_{ij}^k = 2^{i}\times2^{j+k^2}$. But since I am unsure about the notation you are using, I won't attempt to say more.

I understand this completely. So is it that we can think of any tensor as a matrix? I mean even if the there are three or more indices?