# Index Notation question

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1. Sep 21, 2015

### hellomrrobot

I am having trouble showing that $(A_{ik}B_{kj})_{mm} = (A_{ki}B_{jk})_{mm}$

Wouldn't the right side end up having a different outcome? Or can we assume its symmetric?

2. Sep 21, 2015

### Mentz114

It is true for the anti-symmetric part also because the sign change cancels.

3. Sep 21, 2015

### hellomrrobot

anti-symmetric?

4. Sep 21, 2015

### Mentz114

Yes, I think so. A and B can be decomposed into symmetric and anti-symmetric parts.
For the antisymmetric part
$A_{ik}B_{kj}=(-A_{ki})(-B_{jk})$

Is there a 'raised' index here ?

${A_i}^k B_{kj}$

5. Sep 22, 2015

### gill1109

It would help if you would explain your notation.

I suppose that you are using Einstein summation convention. And that when you write (A_{ik}B_{kj}) you mean, the matrix whose i,j element is sum_k (A_{ik}B_{kj}). In other words, the ij element of the matrix AB. Now you want to sum over m the m,m elements of that matrix, so you are talking about trace(AB).

Similarly on the right hand side, replace i and j both by m and m, sum over m and sum over k, and you see that what you have written is just trace(AB).

6. Sep 23, 2015

### stevendaryl

Staff Emeritus
Well, I would say that the right hand side is trace(BA), but that is always equal to trace(AB).

7. Sep 23, 2015

### gill1109

Thank you! You are right.