Tensor Product Explained - Examples Included

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SUMMARY

The tensor product is a mathematical operation that combines multiple tensors into a new tensor. It is denoted as TP = A@B@C@D@..., where "@" represents the product operator. In this context, if A, B, C, and D are tensors, the tensor product TP can be evaluated on 1-forms m, n, o, and p using the relation TP(m,n,o,p) = A(m)@B(n)@C(o)@D(p). This operation results in a tensor of rank 4, as it takes four 1-forms as input.

PREREQUISITES
  • Understanding of tensor algebra
  • Familiarity with 1-forms and their properties
  • Basic knowledge of vector spaces
  • Concept of tensor rank and its implications
NEXT STEPS
  • Study the properties of tensor products in linear algebra
  • Explore examples of tensor products in physics, particularly in quantum mechanics
  • Learn about the applications of tensors in machine learning frameworks like TensorFlow
  • Investigate the relationship between tensor products and multilinear maps
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Ragnar
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Could someone tell me what the tensor product is and give an example?
 
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Ragnar said:
Could someone tell me what the tensor product is and give an example?
The tensor product is a way of formulating a new tensor from other tensors. If you are given the tensors A, B, C, D, ... then the tensor product TP is also a tensor and is represented by the relation

TP = A@B@C@D@...

The "@" is being used for the product operator which is a symbol which actually looks like an x surrounded by a zero. Suppose A, B, C, D are vectors. We feed in the 1-forms m, n, o, p as follows

TP(m,n,o,p) = A(m)@B(n)@C(o)@D(p)

The value of the tensor TP on the one forms is defined in this way. An example is really trivial and you can call the above an example. The tensors don't need to be vectors on the right. They just need to be tensors. Notice that TP is a tensor of rank 4 since it takes in 4 1-forms.

Pete
 

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