Tensors, indices and matrix notation - is there a common convention?

Question

For a tensor named T with two indices, there are four possibilities: $T_{ij}$ , $T_i^{\ j}$, $T^i{\ _j}$ and $T^{ij}$. Is there a common convention as to how these tensors would be represented as matrices, i.e. where the entries would go? Is it the left-right order of the indices that determines which matrix entry is meant, or some other convention? What if the the order of the indices in a mixed tensor is not indicated at all (as in $T_i^j$)? Is it true that, for instance, the component with i=2 and j=3 would go on the second row and the third column in all of the above cases? The books will just say "$F_{μν}$ = [some matrix]", and you don't know which is which.

Below is an example that is in itself contradictory. To convey the idea that F is antisymmetric, they use two different conventions in the very same line - here it is the order of the Greek subscripts that determines the order.

$$ F_{\mu \nu} = \left( \begin{array}{cccc} 0 & -E_1 & -E_2 & -E_3 \\ E_1 & 0 & B_3 & -B_2 \\ E_2 & -B_3 & 0 & B_1 \\ E_3 & B_2 & -B_1 & 0 \end{array} \right) = -F_{\nu \mu} $$

Answer 1

In my experience, reading the indices left to right and top to bottom, the first index is the row and the second is the column.

Your screenshot from Carroll doesn't have to be contradictory (although it's definitely confusing/doesn't make rigorous sense). You can just imagine he omits a little "$_{\mu \nu}$" on the matrix:

$$F_{\mu \nu}=\Bigg( \cdots \Bigg)_{\mu \nu}=-F_{\nu \mu}$$

now it's a true real number equation.

Answer 2

Your example is an outlier, in my experience (personally, I would have written $(F_{\mu\nu})^T$ instead of $F_{\nu\mu}$). Almost always, it's the order of the indices that determines row vs. column. If someone writes $T^i_j$, then while technically there's no way to tell, I would say that it would be far less confusing to make the upper index label the rows and the lower index label the columns. This is because a mixed tensor can be regarded as a linear transformation between vectors:

$$v^i = T^i_j v^j$$

If we want to express this linear transformation as a multiplication of a matrix by a vector, then $j$ should label the columns, since that's the index that is being contracted with the vector $v^j$.

The bottom line, however, is that 99% of the time there will be a left/row index and a right/column index. Representing a tensor as a matrix with any other convention is confusing and should not be done, unless the author has a very strong reason to do so.