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When reading undergraduate GR literature, I often see that the authors represent tensors ${\eta^\alpha}_{\beta}$, ${\eta^\beta}_{\alpha}$, $\eta_{\alpha \beta}$, $\eta^{\alpha \beta}$ as matrices. I wonder which index labels the row number and which index labels the columns number. Of course if the matrix is symmetric then it doesn't matter, but let's suppose the matrix is not symmetric.

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    $\begingroup$ It's a matter of convention. Most commonly, the leftmost index is the row. $\endgroup$ – Ryan Unger Jan 24 '16 at 5:07
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The convention we pick here will interact with the convention we have for matrix multiplication in the following way:

If we have matrices $A$ and $B$ and we use the usual convention that the matrix multiplication $AB$ multiplies the rows of $A$ with the columns of $B$ then we have either

$$(AB)_{ij}=\sum_kA_{ik}B_{kj}\tag{#}$$

or

$$(AB)_{ij}=\sum_kA_{ki}B_{jk}\tag{*}$$

depending on which convention we use for which index labels the columns and which index labels the rows. To me the expression $(\#)$ is more appealing since the indices being summed over are the two which are closest together. Therefore we should chose that in $A_{ij}$ it is $i$ that tells you the row and $j$ that tells you the column.

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