Hey milad.
If something is symmetric then it means that S[i,k] = S[k,i]. Did you try and expand the summation in an explicit way and not in the shorthand Einstein notation?
hello
I am new to the forum so I don't know if I am posting in the right section
I want to prove the following equality in index notation :
S[i,k]S[k,j]S[i,l]S[l,j]=1/2*S[i,j]S[i,j]S[k,l]S[k,l] .
S is a symmetric second-order tensor with property S[i,i] = 0 .
I have been struggling with this for a week ,any help is appreciated, thanks in advance
I expanded the equation using matlab and I got :
S[i,k]S[k,j]S[i,l]S[l,j]-1/2*S[i,j]S[i,j]S[k,l]S[k,l]= s[i,i](~)
which means if we apply the property s[i,i]= 0 and symmetry they are equal .
I also checked this numerically for over 10 different set numbers and if you have symmetry and S[i,i]=0 ,then they are equal .
I meant I checked the expression numerically ,I assigned different sets of numbers to elements of the tensor S , and checked the answers. two sides of the equation are equal if the values assigned to S have those two properties(symmetry , S[i,i]=0 ) .
I solved this and wrote it up in Word because of all the indices. I did a copy paste in Latex help and everything copied correctly. When I tried to put it in this thread, none of the subscripts and superscripts copied over and to do it all manually is a nightmare. So I copied the solution to Latex help and give the link here.
Sol to Einstin indicial notation problem
I would appreciate it if someone put the solution in this post directly. OK, did copy paste from Latex post and that worked.
OP: I want to prove the following equality in index notation :
S[i,k]S[k,j]S[i,l]S[l,j]=1/2*S[i,j]S[i,j]S[k,l]S[k,l] .
let S =|s_{ij}| 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})