Tensor Product Explained - Examples Included

The tensor product is a way of combining multiple tensors to create a new tensor. This is represented by the symbol "@" and is used for the product operator. An example of the tensor product is when we feed in 1-forms to the tensors A, B, C, and D, represented by m, n, o, and p respectively. The resulting tensor, TP, has a rank of 4 since it takes in 4 1-forms.
  • #1
Ragnar
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Could someone tell me what the tensor product is and give an example?
 
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  • #2
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
 
  • #3


The tensor product is a mathematical operation that combines two vector spaces to create a new, larger vector space. It is denoted by the symbol ⊗ and is used in various fields of mathematics, such as linear algebra and functional analysis.

One example of the tensor product is the cross product in three-dimensional Euclidean space. If we have two vectors, a = (a1, a2, a3) and b = (b1, b2, b3), the cross product a ⊗ b is defined as:

a ⊗ b = (a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1)

Another example is the tensor product of two matrices. If we have two matrices A = [a ij] and B = [b kl], the tensor product A ⊗ B is defined as:

A ⊗ B = [a ij * B] = [a11B, a12B, ..., a1nB; a21B, a22B, ..., a2nB; ...; am1B, am2B, ..., amnB]

The resulting matrix will have mn rows and nl columns, where m and n are the dimensions of A and l is the number of columns in B.

In summary, the tensor product is a powerful mathematical tool that allows us to combine vector spaces and perform various operations on them. It has numerous applications in physics, engineering, and computer science, making it an essential concept to understand in mathematics.
 

What is a tensor product?

A tensor product is a mathematical operation that combines two or more tensors to create a new tensor. It is used in linear algebra and is often denoted by the symbol ⊗ (a circle with a cross inside).

How is a tensor product calculated?

To calculate a tensor product, you multiply the components of the first tensor by the components of the second tensor. The resulting tensor will have dimensions equal to the product of the dimensions of the two original tensors.

What are some examples of tensor products?

Some examples of tensor products include the cross product and dot product in vector calculus, and the Kronecker product in matrix algebra.

What are the applications of tensor products?

Tensor products are used in a variety of fields, including physics, engineering, and computer science. They are particularly useful in describing physical phenomena that involve multiple dimensions or quantities, such as forces acting on an object or the properties of a material.

What is the significance of tensor products in quantum mechanics?

In quantum mechanics, tensor products are used to describe the state of a system composed of multiple particles. This allows for a more accurate and comprehensive understanding of quantum systems and their behavior.

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