# Homework Help: Orthogonal change of basis preserves symmetry

1. Sep 28, 2011

### boyboy400

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. Sep 28, 2011

### CompuChip

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

Apply that to S' to show that it is symmetric

3. Sep 28, 2011

### boyboy400

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 ...

4. Sep 28, 2011

### CompuChip

If Sij is the given matrix, then what is Sji called?