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What is the difference between the these two tensors? $$A_{~~i}^{j} \text{ and } A_{i}^{~~j} $$ In my lecturers notes he states that $A^{~~i}_{j}=(A^T)^{i}_{~~j}$. Why is it this and not $A^{~~i}_{j}=(A^T)^{~~i}_{j}$ ? Thank you.

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Thank you for your answers. What i was missing is that the order of the indices from left to right, regardless of their upper or lower position, tells you which is the row and which is the column (for rank 2 case). For some reason I was thinking that the upper index is always row and lower is always column. And if both were upper or both lower, then it would be left is row, right is column. Some times I think the lecturers are so comfortable with certain conventions that they forget to tell the student or assume it is as natural to us as it is to them.

Regardless I could have worked it out by expanding it all out, which I did after writing this post. Sorry for wasting your time.

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Well if you are talking about a symmetric tensor then the quantities are equivalent. But if they are not then it is a similar process to taking the transpose of a matrix.

Conventionally if we have a rank-2 Tensor given by $A^i \;_j$ then the leftmost index represents the rows of a matrix and the columns represented by the remaining index. Hence the transpose is given by $(A^T)^i\;_j=A_j\;^i$. It is equivalent to swapping rows and columns.

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