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?
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 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 :
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 :
let S =|sij| and Sij ≡ sij
S4ii = ½ (S2jj)(S2kk) *
S is symmetric so there is a coordinate system in which S is diagonal with real components. In this coordinate system:
2dim proof (3dim is the same with more algebra. You can’t get beyond this point with summation convention):
s11 + s22 = 0, sum of diagonal elements is a tensor invariant
s112 + s222 = -2s11s22
(s112 + s222) (s112 + s222) = 4s112s222
s114 + s224 = 2s112s122 = ½ (s112 + s222) (s112 + s222)
S4ii = ½ (Sjj)(Skk)
Proof holds in any coordinate system because contraction of a tensor is a tensor.
* sikskjsilslj = S2ijS2ij = S4ii
(AijAjk = A2ik, AijAji = A2ii)