Orthogonal change of basis preserves symmetry

[boyboy400]

Homework Statement

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.

Homework Equations

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

 

[CompuChip]

Well If S is symmetric then Sij=Sji

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

Apply that to S' to show that it is symmetric

 

[boyboy400]

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

Apply that to S' to show that it is symmetric

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 ...

 

[CompuChip]

If S_ij is the given matrix, then what is S_ji called?
You have already used it in your "attempt at a solution".