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In regards to the relation $$S=S_{ij}u_iu_j$$ I've been told that the development leading to this relation is as follows, "To define $S$, you pick a unit vector ($u_i$) and contract it with both indices of $S_{ij}$."

For $i,j=1,2,3$, the (desired) result would be $$S=S_{11}u_1^2+S_{12}u_1u_2+S_{13}u_1u_3+S_{21}u_2u_1+S_{22}u_2^2+S_{23}u_2u_3+S_{31}u_3u_1+S_{32}u_3u_2+S_{33}u_3^2$$

Would it be equivalent and/or correct to say the following as an alternative?: "To define $S$, you pick a unit vector ($u_i$) and multiply it by $S_{ij}$."

I only get the notion of what is meant by "contracting" the vector with the indices of $S_{ij}$, but does the term "contract" have a formal definition or meaning? Is it synonymous with multiplication?

Armadillo
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    How do you define multiplication? Your quantity is $u\cdot Su$ in terms of the dot product & matrix multiplication. (See here for why I didn't just write $u^TSu$.) – J.G. Mar 31 '22 at 15:24
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    "Contracting" two dummy indices refers to making them equal, turning it into a sum. "Multiplication" here is ambiguous, but for me 'multiplying' $S_{ij}$ and $u_i$ would be vector-matrix multiplication $S_{ij} u_j$, which isn't the same. – FlipTack Mar 31 '22 at 15:26
  • @FlipTack (+1) Can you clarify what you mean by making the dummy indices equal? My confusion is this: if we let $S_{ij}=S_{ji}$, we have after performing the summations and collecting terms: $S=S_{11}u_1^2+S_{22}u_2^2+S_{33}u_3^2+2S_{12}u_1u_2+2S_{23}u_2u_3+2S_{13}u_1u_3$ As seen, the first three terms are from the dummy indices $i$ and $j$ being equal. However, the last three terms the dummy indices are not equal. So I am failing to understand what is meant by making the indices equal based on how I "view" these last three terms where $i$ is not equal to $j$. – Armadillo Mar 31 '22 at 15:54

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A contraction is a sum of certain components of a tensor similar to the operation of taking the trace of a matrix. For example if you take the expression $S_{i, j} u_k u_\ell$ with 4 indices, you can contract it by summing all the terms where $i = k$ and $j = \ell$ \begin{equation} S_{i, j} u_i u_j := \sum_i \sum_j S_{i, j} u_i u_j \end{equation} The convention that repeated indices means summation is called Einstein's convention.

With this convention, the trace of the matrix could be denoted $S_{i, i}$.

Gribouillis
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  • (+1) So if my desired result is for $S=S_{11}u_1^2+S_{12}u_1u_2+S_{13}u_1u_3+S_{21}u_2u_1+S_{22}u_2^2+S_{23}u_2u_3+S_{31}u_3u_1+S_{32}u_3u_2+S_{33}u_3^2$, would this be in agreement with the definition of contraction as used in my post? – Armadillo Mar 31 '22 at 16:25
  • Yes I think so. This sum is exactly the result of contracting the order 4 tensor $S_{i, j} u_k u_\ell$ as explained above. – Gribouillis Mar 31 '22 at 16:42
  • So “contraction” or the act of “contracting” is essentially the operation of (the application of) the Einstein summation convention. – Armadillo Apr 01 '22 at 13:49
  • Essentially yes, however tensors are usually defined in a more abstract way than just a collection of numbers. In the same way that matrices represent linear maps or quadratic forms in a certain basis, tensors can represent multilinear maps or forms. Tensors become objects independent of a specific basis and «contraction» also acquires a more abstract sense – Gribouillis Apr 01 '22 at 16:51