Undergrad Symbol for matrix representative of a tensor

[doktorglas]

TL;DR

Searching for a symbol to be used instead of "equals" between a tensor and a matrix.

Hi,

I'm simply searching for some standard symbol (in place of an equals sign) to indicate that a matrix is a representative of a tensor in some given basis. Is there any standardized symbol like this, or how is this usually written in literature?

E.g. say we have a O(1,1) metric tensor g_μν which can be represented in matrix form as diag(1,-1) or, in another basis, as off-diag(1,1), I would not want to write g_μν = diag(1,-1), since the matrix is not the tensor but merely represents the tensor in the specific basis in question. Even though I suppose that it is indeed often written like this, it appears to be an abuse of notation.

Many thanks in advance

 

[fresh_42]

You will have to explain it anyway, so you can take whatever you want. Here are some examples I would choose from:
$$
\triangleq \, \qquad \,\stackrel{1:1}{\longleftrightarrow}\, \qquad \,\circeq\, \qquad \,\bumpeq\, \qquad \,\backsimeq\, \qquad \,\sim\, \qquad \,\fallingdotseq
$$
Another possibility is writing the tensor consequently as $\sum u_\rho \otimes v_\rho$ without defining a basis for $u_\rho ,v_\rho$ and use matrices only after choosing a basis.

 

[doktorglas]

You will have to explain it anyway, so you can take whatever you want. Here are some examples I would choose from:
$$
\triangleq \, \qquad \,\stackrel{1:1}{\longleftrightarrow}\, \qquad \,\circeq\, \qquad \,\bumpeq\, \qquad \,\backsimeq\, \qquad \,\sim\, \qquad \,\fallingdotseq
$$
Another possibility is writing the tensor consequently as $\sum u_\rho \otimes v_\rho$ without defining a basis for $u_\rho ,v_\rho$ and use matrices only after choosing a basis.

Great reply, thank you! I was actually thinking something like an equals sign with a dot above, so I might have seen this somewhere.

Just out of curiosity: Could you point to any references where some of these various alternatives are used? Particularly the $\stackrel{1:1}{\longleftrightarrow}$. What would be the significance of the "1:1" there?

 

[fresh_42]

Just out of curiosity: Could you point to any references where some of these various alternatives are used?

Not without searching for them. Authors are usually far less cautious and do not distinguish between vectors and their coordinates or assume that the basis is clear. Many of the threads in the linear algebra forum are based on this negligence, especially if more than one basis are involved.

Particularly the $\stackrel{1:1}{\longleftrightarrow}$. What would be the significance of the "1:1" there?

Well, it is still a one-to-one correspondence. You could as well introduce a vector space isomorphism
$$
\varphi \, : \,V\otimes W \longrightarrow \mathbb{M}_{n\times m}(\mathbb{F})
$$
and write the matrices as images under $\varphi$ but I think it would only unnecessarily complicate notation. The difficulty is always the same: Did you transform the vectors or the bases? Does the object rotate or the coordinate system? If you want to be clear, you could also write the matrices as
$$
\begin{pmatrix} a&b\\c&d \end{pmatrix}_{\mathcal{B}}
$$
where you defined the basis $\mathcal{B}$ beforehand.

 

[haushofer]

TL;DR Summary: Searching for a symbol to be used instead of "equals" between a tensor and a matrix.

Hi,

I'm simply searching for some standard symbol (in place of an equals sign) to indicate that a matrix is a representative of a tensor in some given basis. Is there any standardized symbol like this, or how is this usually written in literature?

E.g. say we have a O(1,1) metric tensor g_μν which can be represented in matrix form as diag(1,-1) or, in another basis, as off-diag(1,1), I would not want to write g_μν = diag(1,-1), since the matrix is not the tensor but merely represents the tensor in the specific basis in question. Even though I suppose that it is indeed often written like this, it appears to be an abuse of notation.

Many thanks in advance

The $g_{\mu\nu}$ also does not represent the tensor, but the tensor components in some basis (see e.g. Wald's index-free notation). So I don't see the issue here.

 

[doktorglas]

The $g_{\mu\nu}$ also does not represent the tensor, but the tensor components in some basis (see e.g. Wald's index-free notation). So I don't see the issue here.

I would argue that the whole point of the tensor index notation $g_{\mu\nu}$ is exactly the fact that we can write tensor equations without choosing any specific basis -- we only need the index structure of the constituent tensors. So, the symbol $g_{\mu\nu}$ is more of a placeholder for the components than the components themselves. Or did I misinterpret your reply?