(Index Notation) Summing a product of 3 numbers

[throneoo]

I have just begun reading about Einstein's summation convention and it got me thinking..
Is it possible to represent ∑a_ib_ic_i with index notation? Since we are only restricted to use an index twice at most I don't think it's possible to construct it using the standard tensors (Levi Cevita and Kronecker Delta). Levi Cevita doesn't work because it's only non-zero when the indices are all different and Kronecker Delta only connects two tensors, leaving the third one behind. It becomes clearer if I think of them in terms of vectors and matrices.

 

[stevendaryl]

Well, you can introduce a matrix $A_{ijk}$ such that $A_{iii} = 1$ and $A_{ijk} = 0$ if any of the indices differ. Then $\sum_i a_i b_i c_i = A_{ijk} a^i b^j c^k$. The matrix $A_{ijk}$ might not be a very interesting matrix, mathematically.

 

[throneoo]

Well, you can introduce a matrix $A_{ijk}$ such that $A_{iii} = 1$ and $A_{ijk} = 0$ if any of the indices differ. Then $\sum_i a_i b_i c_i = A_{ijk} a^i b^j c^k$. The matrix $A_{ijk}$ might not be a very interesting matrix, mathematically.

That's exactly what I thought. I suppose this ad hoc matrix is not common because this type of summation rarely appears in Physics

 

[stevendaryl]

That's exactly what I thought. I suppose this ad hoc matrix is not common because this type of summation rarely appears in Physics

Well, the most common use of the Einstein summation convention is for manipulations of vectors and tensors. For those uses, one-dimensional matrices such as $a^i$, written with a raised index, corresponds to a vector, and $b_j$, with a lowered index, corresponds to a covector (there is a geometric distinction between vectors and covectors, even though people often ignore the distinction when using Cartesian coordinates for the components). Something with more than one index is a tensor. For example, $g_{ij}$ is the metric tensor, which is used to compute the length of a vector:

$|\vec{V}| = \sqrt{g_{ij} V^i V^j}$

But not every matrix of numbers corresponds to a tensor. The reason why is because tensors transform when you change coordinate systems (for example, changing from Cartesian to polar coordinates). If you define $A_{ijk}$ so that in any coordinate system, $A_{iii} =1$ and $A_{ijk} = 0$ when any two indices are different, then $A$ will not be a tensor.

 

[Orodruin]

Well, you can introduce a matrix $A_{ijk}$ such that $A_{iii} = 1$ and $A_{ijk} = 0$ if any of the indices differ. Then $\sum_i a_i b_i c_i = A_{ijk} a^i b^j c^k$. The matrix $A_{ijk}$ might not be a very interesting matrix, mathematically.

I would not call $A_{ijk}$ a matrix. Something represented by a matrix generally has two indices (or one in the case of row or column matrices). This would be some sort of multidimensional matrix.

Seen as a tensor, this object would not be an isotropic tensor, i.e., it would have different components in another frame, unlike the Kronecker delta or the permutation symbol (seen as a Cartesian pseudo-tensor). In the same fashion $\sum_i a^i b^i c^i$ is not an invariant under coordinate transformations.