How Do Indices Affect Tensor Notation?

[spaghetti3451]

I have learned that there is a difference between the tensors ${T^{\mu}}_{\nu}$ and ${T_{\nu}}^{\mu}$.

Does the upper index denote the rows and the lower index the columns?

 

[jedishrfu]

They are not the same. However they are both mixed tensors. You read more about them here:

https://en.wikipedia.org/wiki/Mixed_tensor

The upper index doesn't denote a row or column similarly for the lower index, instead they denote whether its a covariant (lower) index or a contravariant (upper) index.

 

[spaghetti3451]

I get it: the two important properties of the indices of a tensor are its order in the list of indices and its contravariance/covariance.

For example, the tensors ${T^{\mu}}_{\nu}$ and ${T_{\nu}}^{\mu}$ differ from from each other (even though the index $\mu$ is contravariant in both cases and the index $\nu$ is covariant in both cases) since the order of the indices is different. Am I correct?

 

[jedishrfu]

The u,v naming doesn't really matter as you could have written them as:

${T^{\mu}}_{\nu}$ and ${T_{\mu}}^{\nu}$

What's important is the upper and lower order ie $\mu$, the first index is upper and $\nu$, the second is lower for the first mixed tensor ${T^{\mu}}_{\nu}$.

 

[spaghetti3451]

I am still wondering, though, how to interpret the indices of ${T^{\mu}}_{\nu}$ and ${T_{\nu}}^{\mu}$ in terms of rows and columns.

For example, by convention, the vector $A^{\mu}$ is a column vector, so that the index $\mu$ denotes the row number of the vector.

On the other hand, the dual vector $A_{\mu}$ is a row vector, so that the index $\mu$ now denotes the column number of the vector.

Coming back to ${T^{\mu}}_{\nu}$, does $\mu$, being the contravariant index, denote the row number and $\nu$, being the covariant index, denote the column number of the matrix?

What role does the upper and lower order of the indices serve in terms of the interpretation as matrix elements?

 

[jedishrfu]

I think visually you can lay the tensor out like a matrix with rows and columns. However people tend to use the indices alone and not worry about comparing it to a matrix in that way.

Basically, you don't want to lose sight of the covariant/contravariant nature of each indice.

Here's a writeup on tensor notation where you can see that they use matrix notation for some covariant tensors:

http://www.continuummechanics.org/cm/tensornotationbasic.html

Perhaps @Mark44 can add something here too.

 

[WWGD]

I am still wondering, though, how to interpret the indices of ${T^{\mu}}_{\nu}$ and ${T_{\nu}}^{\mu}$ in terms of rows and columns.

For example, by convention, the vector $A^{\mu}$ is a column vector, so that the index $\mu$ denotes the row number of the vector.

On the other hand, the dual vector $A_{\mu}$ is a row vector, so that the index $\mu$ now denotes the column number of the vector.

Coming back to ${T^{\mu}}_{\nu}$, does $\mu$, being the contravariant index, denote the row number and $\nu$, being the covariant index, denote the column number of the matrix?

What role does the upper and lower order of the indices serve in terms of the interpretation as matrix elements?

I would say a clearer perspective is the one taken from the Wiki article: that an (m+n)-tensor acts on m vectors and n covectors (meaning differential forms), and it is multilinear, i.e., linear on each variable.

 

[spaghetti3451]

I think visually you can lay the tensor out like a matrix with rows and columns. However people tend to use the indices alone and not worry about comparing it to a matrix in that way.

Basically, you don't want to lose sight of the covariant/contravariant nature of each indice.

Here's a writeup on tensor notation where you can see that they use matrix notation for some covariant tensors:

http://www.continuummechanics.org/cm/tensornotationbasic.html

Perhaps @Mark44 can add something here too.

Thanks a lot!