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


    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:


    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, Mar 7, 2008
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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, Mar 10, 2008
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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,
    andrea.panizza75, Mar 12, 2008
  4. deltaquattro

    deltaquattro Guest


    Hi, Glynne,

    how do you prove this? thanks,

    best regards,

    deltaquattro, Mar 13, 2008
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