I'm reading this introduction to tensors: https://arxiv.org/pdf/math/0403252.pdf, specifically rules concerning summation convention (ref. page 13):

Rule 1. In correctly written tensorial formulas free indices are written on the same level (upper or lower) in both sides of the equality. Each free index has only one entry in each side of the equality.

Rule 2. In correctly written tensorial formulas each summation index should have exactly two entries: one upper entry and one lower entry.

Rule 3. For any double indexed array with indices on the same level (both upper or both lower) the first index is a row number, while the second index is a column number. If indices are on different levels (one upper and one lower), then the upper index is a row number, while lower one is a column number.

I have a doubt on applying these rules to matrix multiplication though. Let $A$ and $B$ be matrices and let's represent their elements as $A_{ij}$ and $B^{jk}$. If $C=AB$, then

$$C_i^k = A_{ij}B^{jk}$$

where $j$ is summed over. But on the LHS, $k$ clearly represents the column index, and $i$ the row index. You can even check it yourself by considering $A$ and $B$ as $2\times 2$ matrices. According to rule 3 though, it's supposed to be the opposite since $i$ is the subscript and $k$ the superscript.

Is there a way to resolve the inconsistency between the $3$ rules here, or am I missing something? Because if this convention doesn't even apply to something as simple as matrix multiplication, then it seems pretty useless.


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