skew-symmetric) tensor µ2Aand a symmetric tensor s2S with t=µ+s. Therefore, T= AS : (2) This decomposition is trivially an orthogonal direct sum for the Frobenius inner product h:;:i F due to the fact that the product of an antisymmetric matrix and a symmetric matrix is traceless, and thus their inner product vanishes. Note that
Dec 15, 2000 Transverse traceless gauge - Universe in Problems From now on we consider only vacuum solutions. Suppose we use the Lorenz gauge. As shown above, we still have the freedom of coordinate transformations with $\square \xi^\mu =0$, … Traceless tensors and the symmetric group - ScienceDirect
An Evaluation of Glyph Perception for Real Symmetric
MECHANICAL PROPERTIES OF THE ELECTROMAGNETIC FIELD
We have introduced two 3-scalar fields (x, ) and (x, ), one 3-vector field w(x, ) = w i e i, and one symmetric, traceless second-rank 3-tensor field h(x, ) = h ij e i e j.No generality is lost by making h ij traceless since any trace part can be put into .The factors of 2 and signs have been chosen to simplify later expressions.
Jun 01, 1970 · A tensor is symmetric if its components are unaltered by an interchange of any pair of their indices. A traceless symmetric tensor of order m has 2w+ 1 indepen- dent components, and corresponds to a (2m + 1)-dimensional irreducible representa- tion of the proper orthogonal group in three dimensions. I understand how to create a traceless symmetric tensor, like $$ \hat{X}_{ij} = X_{ij} - \frac{1}{N}\delta_{ij}X_{hh} $$ with Einstein convention of summing over repeated indices. (By the way, I'm following here the book "Group Theory in a Nutshell for Physicists", by A. Zee).