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Inverse of a matrix

  1. Apr 6, 2014 #1
    is it true that [itex]\frac{1}{g_{ab}}=g^{ba}[/itex]? I am a bit confused by the index notation. I especially wonder about the inversion of the indices. Could somebody clarify this please?
     
  2. jcsd
  3. Apr 6, 2014 #2
    No, that's not true. That would be the matrix with reciprocal entries, which is obviously not the inverse.

    It would take me a while to explain the index notation and lowering and raising indices (and some Latex work), which I am not feeling up to right now.
     
  4. Apr 6, 2014 #3

    Fredrik

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    ##g^{ab}## is the number on row a, column b of the inverse of the matrix that has ##g_{ab}## on row a, column b.

    It's not true in general that if A is an invertible matrix, then ##(A^{-1})_{ij}=1/A_{ji}##. Even when A is diagonal, it's only true for the numbers on the diagonal.
     
  5. Apr 6, 2014 #4

    HallsofIvy

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    IF [itex]g^{ij}[/itex] is intended as the fundamental metric tensor, [itex]ds^2= g^{ij}dx_idx_j[/itex], then it is true that [itex]g_{ij}= (g^{ij})^{-1}[/itex] but, again, that is NOT the same as [itex]\frac{1}{g_{ij}}[/itex].
     
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