EDIT: This was just a test to see if superscripts and subscripts would carry over from Word. They did here, but not in original thread. Was able to correct problem in original thread which is at:
Einstein indicial notation problem
let S =|sij| and S_{ij} ≡ s_{ij}
Prove:
S^{4}_{ii} = ½ (S^{2}_{jj})(S^{2}_{kk}) *
S is symmetric so there is a coordinate system in which S is diagonal with real components. In this coordinate system:
S=diag(s_{11},s_{22},s_{33})
S^{2}=diag(s_{11}^{2},s_{22}^{2},s_{33}^{2})
S^{4}=diag(s_{11}^{4},s_{22}^{4},s_{33}^{4})
2dim proof (3dim is the same with more algebra. You can’t get beyond this point with summation convention):
s_{11} + s_{22} = 0, sum of diagonal elements is a tensor invariant
s_{11}^{2} + s_{22}^{2} = -2s_{11}s_{22}
(s_{11}^{2} + s_{22}^{2}) (s_{11}^{2} + s_{22}^{2}) = 4s_{11}^{2}s_{22}^{2}
s_{11}^{4} + s_{22}^{4} = 2s_{11}^{2}s_{12}^{2} = ½ (s_{11}^{2} + s_{22}^{2}) (s_{11}^{2} + s_{22}^{2})
S^{4}_{ii} = ½ (S_{jj})(S_{kk})
Proof holds in any coordinate system because contraction of a tensor is a tensor.
* s_{ik}s_{kj}s_{il}s_{lj} = S^{2}_{ij}S^{2}_{ij} = S^{4}_{ii}
(A_{ij}A_{jk} = A^{2}_{ik}, A_{ij}A_{ji} = A^{2}_{ii})
EDIT Why was I able to copy this to Latex Help from Word but not to post in thread?