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

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

1. ### deltaquattroGuest

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

2. ### ~GlynneGuest

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

3. ### andrea.panizza75Guest

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
4. ### deltaquattroGuest

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

Hi, Glynne,

how do you prove this? thanks,

best regards,

deltaquattro

deltaquattro, Mar 13, 2008