Index notation of matrix tranpose

  • Thread starter birulami
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  • #1
birulami
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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?
 

Answers and Replies

  • #2
WannabeNewton
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No, what he says is perfectly fine: he writes ##\Lambda^{\mu}{}{}_{\sigma} = (\Lambda^T)_{\sigma}{}{}^{\mu}## which is not what you wrote above.
 
  • #3
birulami
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And you are telling me now that this glitch in the typography of the slightly out of place indexes is the crucial point?:cry:

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

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.

[tex]A_\lambda{}^\mu = g_{\lambda\nu} \, A^{\nu\mu} = g_{\lambda\nu} \, g^{\mu\sigma} \, A^\nu{}_\sigma[/tex]
 

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