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Orthogonal change of basis preserves symmetry

  1. Sep 28, 2011 #1
    1. The problem statement, all variables and given/known data

    Prove that symmetric and antisymmetric matrices remain symmetric and antisymmetric, respectively, under any orthogonal coordinate transformation (orthogonal change of basis):

    Directly using the definitions of symmetric and antisymmetric matrices and using the orthogonal transformation rules without reference to components.


    2. Relevant equations



    3. The attempt at a solution

    Well If S is symmetric then Sij=Sji
    and for any u and v in the R space we have u.Sv=Transpose(S)u.v=Su.v
    and S under any change of basis would be S'=QSTranspose[Q]

    but I don't know how to go further...I really appreciate if anyone can help me out with this...I just have a few hours left :(
    Thank you so much
     
  2. jcsd
  3. Sep 28, 2011 #2

    CompuChip

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    There is a coordinate-independent formulation of that, in terms of the transpose.

    Apply that to S' to show that it is symmetric
     
  4. Sep 28, 2011 #3
    Thanks for the response but I guess what you refer to requires using indices and Einstein Summation stuff .. I'm only supposed to use the definitions ... Maybe my formula Sij=Sji was misleading or maybe it was not...
    Could you be a little more specific please? Does that coordinate independent formula have a particular name or something? I can't figure out what you are referring to ...
     
  5. Sep 28, 2011 #4

    CompuChip

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    If Sij is the given matrix, then what is Sji called?
    You have already used it in your "attempt at a solution".
     
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