Proving Tensor Identity | Troubleshooting with Einstein's Summation Convention

In summary, the conversation discusses a tensor identity in Einstein's summation convention. The identity is shown to be equal if a_{ij} is symmetric, but the reason for this symmetry is not clear. The conversation also provides a proof for the identity using index gymnastics.
  • #1
jvicens
1,422
2
I'm having some trouble to prove the following tensor identity shown below in Einstein's summation convention:

[tex](a_{ij}+a_{ji})x_{i}x_{j}=2a_{ij}x_{i}x_{j}[/tex]

I expanded the terms but when I did group them I didn't get the identity. The only way I could get the identity is if

[tex]a_{ij}=a_{ji}[/tex]

and I don't see a reason why this would be so.

Obviously I'm missing something. Can somebody tell what is it that I'm doing wrong?
 
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  • #2
It's true that [tex]a_{ij}x_{i}x_{j} = a_{ji}x_{i}x_{j}[/tex]. Putting in the summation signs may help you to see this.
 
Last edited:
  • #3
To elaborate on Lonewolf's reply...
[tex]
\begin{align*}
a_{ij}x_{i}x_{j}
&= a_{ji}x_{j}x_{i}&&\text{relabel repeated [dummy] indices}\\
&= a_{ji}x_{i}x_{j}&&\text{reorder the writing of the x factors}\\
\end{align*}
[/tex]

You'll learn that [itex]\frac{1}{2}(a_{ij} + a_{ji})[/itex]
is called "the symmetric part of [itex] a_{ij} [/itex]", and is written as [itex] a_{(ij)} [/itex].
Hence, your identity can be written
[tex] 2a_{(ij)}x_{i}x_{j} =2a_{ij}x_{i}x_{j}[/tex].

Here is an "index gymnastics" proof, starting with half of your right-hand-side:
[tex]
\begin{align*}
a_{ij}x_{i}x_{j}
&= a_{ij}x_{(i}x_{j)}&&\text{since } x_{i}x_{j} = x_{(i}x_{j)} \\
&= a_{(ij)}x_{i}x_{j}&&\text{since i and j are being symmetrized}
\end{align*}
[/tex]
 
  • #4
Guys, thanks to you help I got it right this time after some thinking and some calculations. I realized one of my problems was the fact that when I was looking at
[tex]a_{ij}=a_{ji}[/tex]
I was thinking of it as when we say
[tex]a=b[/tex]
The way I should look at that expression is not isolated from the rest of the other terms. The important thing, I think, is not the term itself isolated but how the summation terms comes up after the entire summation is expanded.
I can continue now with the next page of my book :smile:
 
Last edited:

1. What is a tensor identity?

A tensor identity is a mathematical equation that expresses a relationship between two or more tensors. It is used to simplify and solve problems in various fields of science, including physics, engineering, and computer science.

2. Why is understanding tensor identities important?

Understanding tensor identities is important because they are the building blocks of tensor calculus, which is a powerful tool for modeling and analyzing complex systems. Tensor identities also play a crucial role in many areas of modern physics, such as general relativity and quantum field theory.

3. How do you prove a tensor identity?

To prove a tensor identity, you must show that both sides of the equation are equal by manipulating the tensors using known identities and properties. This involves using techniques such as index notation, Einstein summation convention, and tensor contraction.

4. Can tensor identities be applied in real-world applications?

Yes, tensor identities have numerous real-world applications. They are used in fields such as robotics, computer vision, machine learning, and signal processing to model and analyze complex systems. They are also essential in areas like structural engineering and fluid mechanics.

5. Are tensor identities difficult to understand?

Tensor identities can be challenging to understand at first, as they require a solid foundation in linear algebra and differential calculus. However, with practice and patience, they can be mastered. It is important to have a good understanding of the fundamental concepts and notation before attempting to learn tensor identities.

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