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GR: simple algebra

  1. Apr 20, 2009 #1
    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.
     
  2. jcsd
  3. Apr 20, 2009 #2

    CompuChip

    User Avatar
    Science Advisor
    Homework Helper

    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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