Index notation of matrix tranpose

[birulami]

Zee writes in Einstein Gravity in a nutshell page 186

"let us define the transpose by $(\Lambda^T)_\sigma^\mu = \Lambda_\sigma^\mu$"

and even emphasizes the position of the indexes. Yes, they are not exchanged! This must be a typo, right?

 

[WannabeNewton]

No, what he says is perfectly fine: he writes $\Lambda^{\mu}{}{}_{\sigma} = (\Lambda^T)_{\sigma}{}{}^{\mu}$ which is not what you wrote above.

 

[birulami]

And you are telling me now that this glitch in the typography of the slightly out of place indexes is the crucial point?

Oh my, I thought I understood why we have upper and lower indexes? But what is the meaning of an index slighly shifted to the right?

 

[DrGreg]

And you are telling me now that this glitch in the typography of the slightly out of place indexes is the crucial point?

Oh my, I thought I understood why we have upper and lower indexes? But what is the meaning of an index slighly shifted to the right?

The left-to-right order of the indices matters: in a matrix representation the first index is the row index and the second index is the column index.

$$A_\lambda{}^\mu = g_{\lambda\nu} \, A^{\nu\mu} = g_{\lambda\nu} \, g^{\mu\sigma} \, A^\nu{}_\sigma$$