What is a Tensor Product and How Does it Relate to Vectors and Matrices?

In summary, the conversation is about understanding a term related to a transformation T, which involves mutually orthogonal directions and a rectangular plane. The person asking the question is unsure about the symmetry of T and asks for help in expanding it. The response provides a link to more information about tensor products, which may aid in understanding the concept.
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ngc_1729
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Hi, can anyone please explain me how to understand this term? I tried to expand it, but seems I may not be right, so can anyone help me with expasion of this rhs term below? T is suppsoed to be symmetric, but when I expand it it doesn't seem to be symmetric, please help.

consider 2 mutually orthogonal directions a1,a2. associated with sides of a rectangular plane whose sides are d1,d2. and this rectangular plane is oriented at an arbitrary angle wrt global x axis.
Now consier a Transformation T as a function of (a1,a2) and (d1,d2) as :
T = [tex]\Sigma[/tex][(1/di)ai X ai] where i =1 to 2 and X is tensor product

when I expanded rhs of the above experssion I got:

T11 = a1 d1/d1 , T12 = a1d2/d1, T21 = a2d1/d2 , T22 = a2d2/d2
am I correct? if I am why is this not symmetric?
 
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Question 1:

What is the definition of tensor product of two vectors?

The tensor product of two vectors is a mathematical operation that combines two vectors to create a new vector space. It is also known as the Kronecker product and is represented by the symbol ⊗.

Question 2:

How is the tensor product of two vectors calculated?

The tensor product of two vectors is calculated by multiplying each element of one vector with every element of the other vector. The resulting vector will have a length equal to the product of the lengths of the original vectors.

Question 3:

What are the properties of tensor product of two vectors?

The main properties of tensor product of two vectors are linearity, associativity, and distributivity. This means that the tensor product is independent of the order of multiplication, can be distributed over vector addition, and follows the rules of scalar multiplication.

Question 4:

What is the difference between tensor product and dot product?

The tensor product and dot product are two different operations on vectors. The dot product results in a scalar value, while the tensor product results in a new vector. Additionally, the dot product measures the similarity or projection of one vector onto another, while the tensor product creates a new vector space.

Question 5:

What are the applications of tensor product in science?

The tensor product has various applications in different fields of science, such as physics, engineering, and computer science. It is used in quantum mechanics to represent the state of a composite system, in signal processing to analyze signals, and in machine learning to create new features from existing ones.

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