Tensor Product of Covariant and Contravariant Vectors

In summary, it is possible to do the tensor product of two contravariant vectors. It is also possible to do the tensor product of two covariant vectors. The tensor product of these two guys is a bilinear function that acts on pairs of tangent vectors. The Tensor product of these two guys is a basis for all the second order tensors.
  • #1
meteor
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It's possible to do the tensor product of two contravariant vectors?
It's possible to do the tensor product of two covariant vectors?
 
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  • #2
Yes, and yes.

- Warren
 
  • #3
Ok, I think it goes this way:
If you want to do the tensor product of two covariant vectors A and B,with its components represented by two row vectors, then you do ATXB, where X denotes matrix product and T denotes transpose, and the resulting matrix is the tensor product
Similarly, to do the tensor product of two contravariant vectors C and D, you do CXDT
Is this correct?
 
  • #4
meteor said:
Ok, I think it goes this way:
If you want to do the tensor product of two covariant vectors A and B,with its components represented by two row vectors, then you do ATXB, where X denotes matrix product and T denotes transpose, and the resulting matrix is the tensor product
Similarly, to do the tensor product of two contravariant vectors C and D, you do CXDT
Is this correct?

As long as the column vector is to the left of the row vector, you'll get it right. Another way to think about it is to let the indices in the product:

xμxν

represent the addresses in the matrix representation. In other words, the component x2x3 is the matrix element in the 2nd row, 3rd column.
 
  • #5
I have just finished a math tensor course, it's really a lot of fun once you get going!
 
  • #6
I presume when you say covariant vectors you mean in the classical backwards terminology.

So a covariant vector is a linear function L on the tangent space, with values which are numbers, and likewise another such guy M is the same.

The tensor product of these two guys is a bilinear function LtensM acting on pairs of tangent vectors in the only sensible way, i.e. (LtensM)(v,w) =L(v)M(w), product of numbers.

Now if e1,...,er is a basis for tangent vectors and h1,...,hr is the dual basis for covariant vectors, where hj has value zero at all ei except hj(ej) = 1,

then the basic tensors (hj)tens(hk) are a basis for all the second order tensors.

Hence we must be able to write out LtensM in terms of these basic tensors.

I.e. there must be doubly indexed family Cjk of numbers such that

LtensM = summation Cjk (hj)tens(hk).

Now to learn what these numbers Cjk are, note that Cjk is the value of the right hand side on the contravariant tensor (ej)tens(ek). So to compute Cjk we just

apply the left hand side to this same tensor.

I.e. we should have Cjk = (LtensM)((ej)tens(ek)) = L(ej)M(ek).

Hence if the covariant vector L was represented by (...aj...), and M was represented by (...bk...), i.e. if L(ej) = aj, and M(ek) = bk, then L tens M is represented by the matrix product

(ajbk) = (aj)T (bk), which seems to agree with what was said above.

Now my argument for the conceptual approach is that although I did not know how to multiply tensors when I started this post, I still seem to have got it right because I knew what a tensor actually is.

Or at least if I got it wrong, you can follow what I did and see where I went wrong.
 

1. What is the tensor product of covariant and contravariant vectors?

The tensor product of covariant and contravariant vectors is a mathematical operation that combines two vectors of different types (covariant and contravariant) to create a new type of vector. This operation is used in tensor calculus to represent the relationship between different coordinate systems.

2. How is the tensor product of covariant and contravariant vectors calculated?

The tensor product of covariant and contravariant vectors is calculated using the outer product, also known as the Kronecker product, of the two vectors. This involves multiplying the components of the covariant vector with the components of the contravariant vector to create a new tensor with a higher dimension.

3. What is the significance of the tensor product in physics?

The tensor product is an essential concept in physics, particularly in the study of general relativity. It is used to describe the curvature of space-time and the behavior of physical quantities under different coordinate systems. The tensor product allows for the representation of physical laws and equations in a coordinate-independent manner.

4. What are the properties of the tensor product of covariant and contravariant vectors?

The tensor product of covariant and contravariant vectors has the following properties: it is bilinear, associative, and distributive. This means that it is linear with respect to both vectors, it follows the associative property, and it can be distributed over addition or subtraction of vectors.

5. How is the tensor product of covariant and contravariant vectors used in machine learning?

In machine learning, the tensor product of covariant and contravariant vectors is used to represent multi-dimensional data. It allows for the efficient manipulation and processing of data with a large number of features. Tensors, which are created using the tensor product, are also used in deep learning models to perform operations such as convolution and pooling.

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