# Tensor contraction

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?

## 1 Answer

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.

• What about the element $C^{11}B_2$? It is in none of the sums. – Samapan Bhadury Oct 20 '17 at 18:06
• @SamapanBhadury: it's not set to zero, $C^{i}$ just doesn't depend on it. – Jerry Schirmer Oct 20 '17 at 18:39