showing scalars are unchanged by rotation

[MathematicalPhysics]

Hello, just hoping someone can give me a hand here.

I have a second-order tensor P, which has components p_{ij} and I want to show that the following scalar quantities are unchanged by rotation:

p_{ii}
p_{ij}p_{ji}
p_{ij}p_{jk}p_{ki}

Now, I know scalars are zero'th order tensors, I know im going to have to use the tensor transformation law, I know I must keep in mind the orthogonality of the rotation matrix and I must use the substitution property.

This is what i've done but im not happy that its valid as a solution to my problem.

The transformation law tells us that {p^'}_{ii} = \alpha_{ia} \alpha_{ib} p_{ab}

If it is isotropic then l.h.s = p_{ii} & r.h.s = \alpha_{ia} \alpha_{ia} by the substitution property. This is equal to p_{ii} by the orthogonality of the rotation matrix.

Im not happy with this, any help is much appreciated! Thanks, Matt.

p.s. this is only the first quantity!

[MathematicalPhysics]

I worked out how to do it now, if anyone wants to know..

p'(ii) = alpha (ia) alpha (ib) p (ab)

p'(ii) = delta (ab) p (ab)

p'(ii) = p (aa)

which in this case can be rewritten to look like

p'(ii) = p(ii)

This implies p(ii) is invariant, or unchanged by rotation.