Proving tensor symmetry under transformation

[Hendrick]

Homework Statement

Using indical notation, prove that a 2nd order symmetric tensor D remains symmetric when transformed into any other coordinate system.

Homework Equations

Tensor law of transformation (2nd order):
D'_{pq} = a_{pr}a_{qs}D_{rs}

The Attempt at a Solution

I think I'm required to prove that D'_{pq} = D'_{qp} (where D is a symmetric 2nd order tensor)


D'_{pq} = a_{pr}a_{qs}D_{rs}
D'^{T}_{pq} = (a_{pr}a_{qs}D_{rs})^T
D'_{qp} = a_{qs}a_{pr}D_{sr} (can someone please explain why when you transpose this, the a's swaps position but the D swaps indices?)

D_{rs} = D_{sr} (as D is symmetric)

=>D'_{pq} = a_{pr}a_{qs}D_{rs} = a_{qs}a_{pr}D_{rs} = D'_{qp}

Thus D'_{pq} = D'_{qp}


Did I prove it correctly?

Thank you

 

[Hendrick]

Did I not write my question properly or does no one know?