Do orthogonal transformations preserve the symmetry of the tensors?

Question

I have the following doubt: Do orthogonal transformations preserve the symmetry of the 2 rank mixed tensors?

It seems logical to me since the symmetry of the tensor needs to be preserved if we change from systems of coordinates (iff the transformation matrix is ortogonal). But Is that correct? or How can I prove it?

Thanks in advance.

Answer 1

A transformation $A^\prime_{ij}=O_{ih}O_{jk}A_{hk}$ with $A_{hk}=A_{kh}$ satisfies$$A^\prime_{ji}=O_{jh}O_{ik}A_{hk}=O_{ik}O_{jh}A_{kh}=A^\prime_{ij}.$$