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Say I have to take the transpose of the rotation matrix $\mathcal{O}^{i'}_i$. Can I write it as $\mathcal{O}^i_{i'}$? Note that this is part of an equation with other tensors that use the indeces $i$ and $i'$, so they are not just dummy indeces.

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  • $\begingroup$ To me, usually taking the "transpose" in tensor calculus involves multiplying by the metric tensor of the appropriate variance. For instance, you would be looking for something like $g_{i'k'}g^{ik}\mathcal O^{k'}{}_{k}$, assuming $g$ is the metric tensor. Whether that's actually what you're after depends entirely on what kind of tensor calculus you're working with, though. Just the fact that you're talking about matrices and transposes probably means your flavor is different from mine. $\endgroup$ – Arthur Oct 15 at 18:27

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