# Summing tensors in mathematica

How do I perform the following summation in mathematica? $$\Sigma_{m=1}^5 e_{ijklm}A^{mn}$$ I have the $e_{ijklm}$ tensor of rank 5 in 5 dimension as a array and $A^{mn}$ as a 5x5 matrix.

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Since the sum goes over the last index of $e$ and the first index of $A$, it is directly done by using Dot:

dim = 5;

e = Array[\[ScriptE], Table[dim, {dim}]];

a = Array[\[ScriptA], Table[dim, {2}]];

c = e.a;


Here I defined the arrays with the appropriate dimensions but suppressed the output because it's too long for five dimensions.

Another interesting alternative for more general sums is what I mentioned in this answer, but it requires version 9:

TensorContract[TensorProduct[e, a], {dim, dim+1}] == c


True

Here the {dim, dim+1} are just the last index of the first factor and the first index of the second factor. The latter can be generalized to sums over say k etc.

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