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Explicit and Implicit Matrix Notation Question

  1. Feb 23, 2010 #1
    Ok so this is a fairly stupid question I'm sure, but I'm not quite clear about the following:

    Given a Lorentz transformation we require the following to hold:

    [tex]g_{\sigma\rho}\Lambda^{\sigma}_{\mu}\Lambda^{\rho}_{\nu} = g_{\mu\nu}[/tex]

    In other notation this is written:

    [tex]\Lambda^{T}g\Lambda = g[/tex]

    where [tex]\Lambda \in O(1,3)[/tex] and g is the Minkowski metric.

    The first use of tensor notation is fine, however I am unsure why in the second expression the first [tex]\Lambda[/tex] is transposed?

    Any help greatly appreciated,
    Alex
     
  2. jcsd
  3. Feb 23, 2010 #2
    In the first sum over sigma, the "rows" of g are summed with the "rows" of lambda. In matrix multiplication, rows are summed with columns. If we write lambda and g as matrices and want to express the same product as a matrix multiplication, then in order for the correct components to be multiplied, we must transpose lambda.
     
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