How to calculate the matrix of a form?

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In summary, the concept of a "matrix of an n-form" does not exist. A 2-form can be represented by a matrix with components of the form ##\omega_{ij}##, but this does not apply to a general n-form with n indices. The screenshot from V.I Arnold's book on Classical mechanics shows a matrix with components ##\omega_{ij}## for a 2-form. This can be written as a wedge product for the dual basis, defined as $$dx^i \wedge dx^j = dx^i \otimes dx^j - dx^j\otimes dx^i$$. From there, you can find the appropriate positions for the components in the matrix.
  • #1
Abhishek11235
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This is screenshot from V.I Arnold's book on Classical mechanics. My question is how do we find matrix of any n-form. Detailed answer please.
Screenshot_2019-02-11-16-28-22.jpeg
 

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  • #2
There is no such thing as a "matrix of an n-form". The point is that a 2-form has components of the form ##\omega_{ij}## with two indices, that can be represented as the components of a matrix. This is not the case for a general ##n##-form, which has components with ##n## indices.
 
  • #3
So,how to find matrix given in screenshot for 2 forms?
Orodruin said:
There is no such thing as a "matrix of an n-form". The point is that a 2-form has components of the form ##\omega_{ij}## with two indices, that can be represented as the components of a matrix. This is not the case for a general ##n##-form, which has components with ##n## indices.
 
  • #4
Its just a matrix containing the components ##\omega_{ij}## in the appropriate positions.
 
  • #5
Orodruin said:
Its just a matrix containing the components ##\omega_{ij}## in the appropriate positions.
Can you explain it in brief? I am new to forms. I forgot where it was written in book.
 
  • #6
Are you familiar on how to write down the components of a rank 2 tensor in a matrix?
 
  • #7
Orodruin said:
Are you familiar on how to write down the components of a rank 2 tensor in a matrix?
Yes
 
  • #8
So the wedge product for the dual basis is defined as
$$
dx^i \wedge dx^j = dx^i \otimes dx^j - dx^j\otimes dx^i.
$$
You can work from there.
 

Related to How to calculate the matrix of a form?

1. What is the matrix of a form?

The matrix of a form is a mathematical representation of a linear transformation between two vector spaces. It is a rectangular array of numbers that describes how the elements of one vector space are transformed into the elements of another vector space.

2. How do you calculate the matrix of a form?

To calculate the matrix of a form, you first need to determine the basis vectors for both the input and output vector spaces. Then, you apply the linear transformation to each basis vector and record the resulting coordinates in the matrix. The columns of the matrix represent the coordinates of the transformed basis vectors in the output vector space.

3. What is the purpose of calculating the matrix of a form?

The matrix of a form is useful because it allows us to easily perform calculations and manipulations on linear transformations. It also helps us understand the properties and behavior of the transformation, such as its invertibility and eigenvalues.

4. Are there any special techniques for calculating the matrix of a form?

Yes, there are several techniques for calculating the matrix of a form, such as using Gaussian elimination or using matrix multiplication. The specific technique will depend on the complexity of the linear transformation and the tools available.

5. Can the matrix of a form be calculated for any type of linear transformation?

Yes, the matrix of a form can be calculated for any linear transformation between two vector spaces. However, for more complex transformations, the calculations may be more involved and may require the use of advanced mathematical techniques.

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