Understanding the Properties of Tensor Products: A Demonstration

In summary, the conversation discusses the product of an antisymmetric tensor A^{rs} and a symmetric tensor B_{rs}, which is demonstrated to be equal to zero. Further, it is noted that interchanging and relabeling an implied double summation is acceptable.
  • #1
Telemachus
835
30
I have to demonstrate that if [tex]A^{rs}[/tex] is an antisymmetric tensor, and [tex]B_{rs}[/tex] is a symmetric tensor, then the product:
[tex]A^{rs}B_{rs}=0[/tex]

So I called the product:
[tex]C^{rs}_{rs}=A^{rs}B_{rs}=-A^{sr}B_{sr}=-C^{rs}_{rs}[/tex]
In the las stem I've changed the indexes, because it doesn't matters which is which, but I'm not sure this is fine (because I think r and s could have have associated differents values in the sum).

Then
[tex]2C^{rs}_{rs}=2A^{rs}B_{rs}=0[/tex]
Is this ok?
 
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  • #2
interchanging and relabeling an implied double summation is okay.
 
  • #3
Thank you xaos :)
 

Related to Understanding the Properties of Tensor Products: A Demonstration

1. What is a tensor product demonstration?

A tensor product demonstration is a way of visually representing the mathematical concept of a tensor product. It involves combining two or more vectors or matrices to create a new, higher-dimensional object.

2. How is a tensor product demonstration useful?

A tensor product demonstration can help to illustrate the properties and applications of tensor products, which are widely used in fields such as physics, engineering, and computer science. It can also aid in understanding abstract mathematical concepts.

3. What tools are needed for a tensor product demonstration?

A tensor product demonstration can be done using physical objects such as blocks or cubes, or with software programs that allow for the manipulation and visualization of vectors and matrices. Other helpful tools may include graph paper and colored markers.

4. Can a tensor product demonstration be done with more than two vectors or matrices?

Yes, a tensor product demonstration can involve any number of vectors or matrices. However, as the number of objects increases, the visual representation may become more complex and harder to interpret.

5. Are there different types of tensor product demonstrations?

Yes, there are different ways to demonstrate tensor products depending on the specific application or context. Some common types include outer product demonstrations, Kronecker product demonstrations, and tensor network demonstrations.

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