Tensor Index notation vs Matrix notation Transpose

Referring to the answer in the following question:

https://physics.stackexchange.com/a/349030/288587

I just cant figure out how to go from:

$$\eta_{\mu\nu} = \Lambda^\alpha_{\;\mu}\Lambda^\beta_{\;\nu}\eta_{\alpha\beta}$$

to:

$$η=Λ^TηΛ$$

I tried to do the following:

\begin{align} \eta_{\mu\nu} &= \Lambda^\alpha_{\;\mu}\Lambda^\beta_{\;\nu}\eta_{\alpha\beta} \\ &= \Lambda^\alpha_{\;\mu}\Lambda_{\alpha\nu} \\ &= \Lambda^\alpha_{\;\mu}\eta_{\alpha\sigma}\Lambda^\sigma_{\;\nu} \\ &= \Lambda^\nu_{\;\mu}\eta_{\nu\mu}\Lambda^\mu_{\;\nu} \end{align}

Implying that the transpose of $$\Lambda^\mu_{\;\nu}$$ is $$\Lambda^\nu_{\;\mu}$$ but, as i know, this is not true, as the transpose of $$\Lambda^\mu_{\;\nu}$$ should be $$\Lambda_{\nu\;}^{\;\mu}$$.

Where have i gone wrong?

• By $\eta$ do you mean simply the trace of it? – DJA Feb 10 at 3:06
• And by $\eta$ i mean the metric tensor – Jbfm Feb 10 at 11:47
• Im sorry, but i still dont see how would $(\Lambda^T)_{\nu}{}^{\mu}$ := $\Lambda^\mu{}_\nu$ would be equal $\Lambda^\nu_{\;\mu}$. So i still cant really understand the conversion from matrix to tensor index notation. – Jbfm Feb 10 at 12:59
• I think i got it!! The fact that both tensors on $\Lambda^\alpha_{\;\mu}\eta_{\alpha\sigma}$ have the summation index on the same slot (e.g. as $\alpha$ is summed over, the "columns" of both $\Lambda^\alpha_{\;\mu}$ and $\eta_{\alpha\sigma}$ are multiplied instead of "row"x"column") .This indicates that $\Lambda^\alpha_{\;\mu}$ is the transpose right? If we had $\eta_{\sigma\alpha}$, then we would see $\Lambda^\sigma_{\;\nu}$ as the transpose, correct? – Jbfm Feb 11 at 11:24