Expression of symmetric and skew-symmetric 3x3 matrices in terms oftheir invariants

Discussion in 'Numerical Analysis' started by deltaquattro, Mar 7, 2008.

  1. deltaquattro

    deltaquattro Guest

    Hi,

    in a code I'm writing I need to evalute Tr(W.(W.S)) where Tr is
    trace, . is the matrix product and W and S are symmetric and skew-
    symmetric part of a 3x3 matrix U:

    W=1/2*(U-U^T)
    S=1/2*(U+U^T)

    I would like to express W and S in terms of their invariants, so that
    I can write a simple expression for Tr(W.(W.S)) instead than writing a
    summation of 27 terms. Is there a way to do this? This is very
    important for and I'll appreciate a lot any help. Thanks,

    best regards,

    deltaquattro
     
    deltaquattro, Mar 7, 2008
    #1
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  2. deltaquattro

    ~Glynne Guest


    Since W is 3x3 and skew, you can effectively treat it as a vector, w.

    Thus a dot product with W becomes a cross product with w, e.g.
    W.b = w^b


    Of particular interest to your problem:
    W.W = w^W
    = ww - (w.w)I

    Which yields:
    tr(W.W.S) = tr(S.ww) - (w.w)tr(S)
    = w.S.w - (w.w)tr(S)


    You can also play games to make it appear more symmetric:
    (w.S.w)tr(I)/3 - (w.I.w)tr(S)

    ~Glynne
     
    ~Glynne, Mar 10, 2008
    #2
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  3. Thanks, Glynne,

    the last one looks similar to the identity I found by expanding the
    summation and simplifying terms. I'll check it out.

    Best regards,
    deltaquattro
     
    andrea.panizza75, Mar 12, 2008
    #3
  4. deltaquattro

    deltaquattro Guest

    ^^^^^^^^^^^^^^^^^^^^^^^

    Hi, Glynne,

    how do you prove this? thanks,

    best regards,

    deltaquattro
     
    deltaquattro, Mar 13, 2008
    #4
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