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I am having some fundamental issues with manipulating Lorentz transformation matrices using index notation. However I'm struggling to pin down what the actual issue is.

Hopefully my misunderstanding can be demonstrated by an example. It can be shown that$$(\Lambda^{-1})^{\mu}_{\,\,\,\nu}=\Lambda_{\nu}^{\,\,\,\mu}$$ If I then write $$(\Lambda^{-1})^{\mu}_{\,\,\,\nu}=(\Lambda^{T})^{\mu}_{\,\,\,\nu}$$ I have seemingly proved that $$\Lambda\Lambda^{T}=I$$ which is not true, given the definition of the Lorentz group. I have seen the equation $$(\Lambda^{T})^{\mu}_{\,\,\,\nu}=\Lambda_{\nu}^{\,\,\,\mu}$$ used in various places, so I can't really see what I've done wrong here.

If somebody could point out where I'm going wrong, and explain why, that would be so helpful! Thanks.

Edit: Possible duplicate doesn't seem to explain why I cant: a) Transpose and swap indices of a LT (eq. 4, used in going from eq. 1 to eq. 2) and/or b) Equate matrices, given equal elements (eq. 3) which are needed to solve my example.

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marked as duplicate by AccidentalFourierTransform, ZeroTheHero, sammy gerbil, Qmechanic Mar 30 '17 at 5:33

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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The matrix transpose is given by $$ (\Lambda^T)^\mu{}_\nu = \Lambda^\nu{}_\mu = \overline\delta^{\mu\beta} \underline\delta_{\nu\alpha} \Lambda^\alpha{}_\beta $$ in terms of the 'Kronecker tensors' from this related answer.

In general, that's different from $$ (\Lambda^{-1})^\mu{}_\nu = \Lambda_\nu{}^\mu = \eta^{\mu\beta} \eta_{\nu\alpha} \Lambda^\alpha{}_\beta $$

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