Index notation of matrix tranpose

1. Oct 13, 2013

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?

2. Oct 13, 2013

WannabeNewton

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

3. Oct 13, 2013

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?

4. Oct 13, 2013

DrGreg

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.

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