Independent components of tensor

In summary: How many have m< l? Knowing those automatically gives an equal number. how many have m= l? You know those are 0. Now, how many terms are there where i< k and m< l? Knowing those automatically gives the others. And you also know that there are 36 "diagonal" terms with i= k and m= l. All together, you can choose [itex]\binom{4}{2}^2[/itex]= 36+ 36= 72 terms. The other 256- 72= 184 terms are not independent.In summary, the number of independent components is reduced from 4^4 =
  • #1
cscott
782
1

Homework Statement



For indices running from 0-3:
[tex]R_{iklm} = -R_{kilm} = -R_{ikml}[/tex]

With the above conditions how do I know the number of independent component is reduced from 4^4 = 256 to 36.


No idea how to figure this out.
 
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  • #2
cscott said:

Homework Statement



For indices running from 0-3:
[tex]R_{iklm} = -R_{kilm} = -R_{ikml}[/tex]

With the above conditions how do I know the number of independent component is reduced from 4^4 = 256 to 36.


No idea how to figure this out.

When in doubt, look at a simpler case. If you know that [tex]R_{ij}= -R{ji}[/tex], then knowing [tex]R_{ij}[/tex] for i< k immediately gives you the value of [tex]R_{ji}[/tex]: Of the 16 possible values of [tex]R_{ij}[/tex], there are 4 with i= j so 16- 4= 12 where i is not equal to j and so 12/2= 6 where i< j. Knowing those 6 tells you the other 6. Further, if i= j then [tex]R_{ii}= -R_{ii}[/tex] so that must be 0: The 4 "diagonal" terms must be 0. You can choose 6 of the possible 16 terms of [tex]Rij[/tex].

Now, for this problem you know that R_{iklm} = -R_{kilm}[/itex]. Of the 44= 256 terms, how many have i< k? Knowing that many automatically gives you an equal number. How many have i= k? You know those are 0.

You also know [tex]R_{iklm}= -R_{ikml}[/tex]. Do the same thing with m and l.
 

1. What are independent components of a tensor?

The independent components of a tensor refer to the components of the tensor that are not related to each other by any linear transformation. In other words, they are the components that cannot be expressed as a linear combination of other components.

2. How are independent components of a tensor determined?

The independent components of a tensor can be determined through a process called tensor decomposition. This involves finding a set of basis tensors that span the space of the original tensor, and then identifying the independent components as the coefficients of these basis tensors.

3. Why are independent components important in tensor analysis?

Independent components are important in tensor analysis because they allow for a more efficient representation of a tensor. By identifying and removing the dependent components, the tensor can be simplified and the underlying relationships between its components can be better understood.

4. Can a tensor have more than one set of independent components?

Yes, a tensor can have multiple sets of independent components. This is because there are different ways to decompose a tensor depending on the chosen basis tensors. However, the number of independent components will always be the same regardless of the chosen basis.

5. How are independent components used in practical applications?

Independent components of a tensor are used in a variety of practical applications, such as signal processing, image recognition, and machine learning. They can help reduce the dimensionality of data, improve the accuracy of models, and identify underlying patterns and relationships within the data.

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