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.

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

 

[sardar]

No, this is not a typo. The index notation of matrix transpose is defined as follows:
The transpose of a matrix A is denoted as A^T, and its elements are given by (A^T)_ij = A_ji. In other words, the rows and columns of the matrix are flipped, and the indexes are not exchanged. This is the standard definition of transpose in mathematics and is consistent with the notation used in physics, as shown in Zee's book. So, there is no typo here, and this notation is correct. It is important to note that this notation is different from the notation used in some programming languages, where the indexes are indeed exchanged. As a scientist, it is crucial to be aware of these notational differences to avoid confusion and ensure accurate communication in our work.