GR: simple algebra

  • Thread starter proguinn
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  • #1
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This isn't homework but since it's a simple algebra thought I suppose it's the most appropriate forum- i'm working through some basic general relativity and just came across a line i think i'm being stupid about:

[tex]
g_{\bar{i}\bar{j}}=\Lambda_{\bar{i}}_{i}\Lambda_{\bar{j}}_{j} g_{ij}
[/tex]

then, it's said that it's obvious that this means:
[tex]
g_{\bar{i}\bar{j}}= (\Lambda^T g \Lambda)_{\bar{i}\bar{j}}
[/tex]

Is there an easy way to show this, other than by inspection?

thanks.
 

Answers and Replies

  • #2
CompuChip
Science Advisor
Homework Helper
4,302
47
If A and B are matrices, then the matrix product is given by
(AB)ik = Aij Bjk
by definition, and the transpose of A is given by
(AT)ij = Aji

From that it easily follows (just write them in the correct order, then swap the indices on one of them creating a transpose).
 

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