# I (Index Notation) Summing a product of 3 numbers

1. Sep 29, 2016

### throneoo

I have just begun reading about Einstein's summation convention and it got me thinking..
Is it possible to represent ∑aibici 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.

Last edited: Sep 29, 2016
2. Sep 29, 2016

### stevendaryl

Staff Emeritus
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.

3. Sep 29, 2016

### throneoo

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

4. Sep 29, 2016

### stevendaryl

Staff Emeritus
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.

5. Sep 29, 2016

### Orodruin

Staff Emeritus
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.