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Suppose, there is a tensor of rank three, given by $C^{ij}_k$ and it is a product of two other tensors. The relation is as follows:

$C^{ij}_k$ = $A^{ij}$$B_k$

This $C^{ij}_k$ has 27 elements in it.

If I contract the indices $j$ and $k$ Then I get the following:

$C^i$ = $A^{ij}$$B_j$

By equating $j$ and $k$ the tensor has been reduced to a rank 1 tensor and has only 3 elements. The process is now similar to ordinary matrix multiplication between a 3X3 and a 3X1 matrices.

I get that by equating two indices, we are selecting only a few of the elements of original $C^{ij}_k$ and summing them up, which eventually represents a vector.

My question is, what happens to the other elements? Do they become zero for some reason? If so, what property makes them zero? If they are not zero, why we are allowed to simply ignore them? what do they represent?

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The other elements are not zero, they are "used up".

$$C^{1} = C^{11}B_{1} + C^{12}B_{2} + C^{13}B_{3}$$

each of the $C^{i}$ is a combination of several elements from the old matrix.

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  • $\begingroup$ What about the element $C^{11}B_2$? It is in none of the sums. $\endgroup$
    – Samapan Bhadury
    Oct 20, 2017 at 18:06
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    $\begingroup$ @SamapanBhadury: it's not set to zero, $C^{i}$ just doesn't depend on it. $\endgroup$
    – Zo the Relativist
    Oct 20, 2017 at 18:39

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