How many independent components does a tensor have?

[MatinSAR]

Homework Statement

How many independent component does a tensor have?

Relevant Equations

Tensor analysis.

It's a 4th-dimensional 4th-rank tensor so at first we have $4^4=256$ components.
According to the book, Given that $R_{iklm}=-R_{ikml}$ 256 components reduces to 96. But I cannot see how.
For one pair of i,k 16 components are dependent. We have 12 pairs of i,k(for $i≠k$ becsuse for i=k tensor's components should be 0) so 256 should reduces to $256-12*16=4*16=64$.Thanks for your help ...

 

[Hill]

We have 16 independent combinations of $i,k$. For each such combination, we have an antisymmetric tensor $R_{..lm}=-R_{..ml}$. ... That gives us $96$ components.

 

[BvU]

We have 12 pairs of i,k

only six pairs: 01 02 03 12 13 23

$\$

 

[MatinSAR]

According to the book, Given that $R_{iklm}=-R_{ikml}$ 256 components reduces to 96. But I cannot see how.
For one pair of i,k 16 components are dependent. We have 12 pairs of i,k(for $i≠k$ becsuse for i=k tensor's components should be 0) so 256 should reduces to $256-12*16=4*16=64$.

A better idea just came in to my mind:
For $m=l$ we have $R_{ikmm}=0$ so $4^3=64$ compenents should be zero. Therefor we have 192 independent components. If I divide this 192 by 2 I'll get the right answer.
For $m≠l$ we have $R_{iklm}=-R_{ikml}$ but how it imply that 192 should be divivded by 2?

 

[MatinSAR]

We have 16 independent combinations of $i,k$. For each such combination, we have an antisymmetric tensor $R_{..lm}=-R_{..ml}$. ... That gives us $96$ components.

The problem is that I cannot see why.

only six pairs: 01 02 03 12 13 23

$\$

Did you remove duplicate combinations? (like 3 2)

 

[Hill]

The problem is that I cannot see why.

The antisymmetric tensor has 6 independent components. That is, 6 for each one of 16 combinations of $i,k$. Total: $16 \times 6=96$.

 

[renormalize]

The problem is that I cannot see why.

Did you remove duplicate combinations? (like 3 2)

That's what independent means: if 32 = -23 there is only one independent component, not two. So only 01, 02, 03, 12, 13 & 23 are independent (and 00, 11, 22, 33 all vanish), so 6 independent components total.

 

[MatinSAR]

The antisymmetric tensor has 6 independent components.

That's what independent means: if 32 = -23 there is only one independent component, not two. So only 01, 02, 03, 12, 13 & 23 are independent (and 00, 11, 22, 33 all vanish), so 6 independent components total.

This is true only for 4th-dimensional 4th-rank tensor? Or every tensor?
What a bout symmetric tensor? Does it have 6 independent components?

 

[Hill]

This is true only for 4th-dimensional 4th-rank tensor?

4-dimensional, rank 2.

What a bout symmetric tensor? Does it have 6 independent components?

No. You can make a list or draw a matrix, and count.

 

[renormalize]

This is true only for 4th-dimensional 4th-rank tensor? Or every tensor?
What a bout symmetric tensor? Does it have 6 independent components?

My count was for an antisymmetric rank-2 tensor in 4 dimensions (for your example, the symmetries of the rank-4 Riemann tensor behave like the product of two independent antisymmetric rank-2 tensors, hence 6 x 6 = 36 independent components total). A symmetric rank-2 tensor in 4-dimensions has 10 components: 00, 01, 02, 03, 11, 12, 13, 22, 23, 33. In other dimensions the counts are different.

 

[MatinSAR]

So I should think more about it. Thank you to everyone.

 

[haruspex]

For an NxN matrix, the number of independent elements is
general: $N^2$
symmetric: $\frac 12N(N+1)$
antisymmetric: $\frac 12N(N-1)$

 

[MatinSAR]

Thank you to everyone ... I've understand the first step.
We have 256 components. For $R_{iklm}$ we know there are 16 combinations of i and k and there are 16 combinations of m and l.
Using equation $R_{iklm}=-R_{ikml}$ we know there are 10 dependent components. So we have $256 - 16(10) = 96$ independent components.

Now using next equation $R_{iklm}=-R_{kilm}$ again there are 96 independent components. But how can I find out How many of these 96 independent components also apply to the previous condition and are now repeated?

 

[MatinSAR]

One more question, Should I use all three equations?

 

[haruspex]

One more question, Should I use all three equations?
View attachment 338924

That's only two equations. Any two imply the third.

 

[MatinSAR]

At first we have 256 independent components, using $R_{iklm}=-R_{ikml}$ 256 reduces to 96.
Then using $R_{iklm}=-R_{kilm}$ 96 reduces to 36. Am I right?

 

[renormalize]

We have 256 components. For $R_{iklm}$ we know there are 16 combinations of i and k and there are 16 combinations of m and l and.
Using equation $R_{iklm}=-R_{ikml}$ we know there are 10 dependent components. So we have $256 - 16(10) = 96$ independent components.

Now using next equation $R_{iklm}=-R_{kilm}$ again there are 96 independent components. But how can I find out How many of these 96 independent components also apply to the previous condition and are now repeated?

Consider the rank-4 product $P_{iklm}=A_{ik}B_{lm}$ of two arbitrary rank-2 tensors $A_{ik}$ and $B_{lm}$. Each tensor $A$ and $B$ has in general $16$ components, so the product $P$ starts with $16\times 16=256$ distinct components. Now impose the two independent antisymmetries:$$P_{iklm}=-P_{kilm}\Rightarrow A_{ik}B_{lm}=-A_{ki}B_{lm}\Rightarrow A\text{ has 6 components}$$$$P_{iklm}=-P_{ikml}\Rightarrow A_{ik}B_{lm}=-A_{ik}B_{ml}\Rightarrow B\text{ has 6 components}$$So the two antisymmetries together give $P=A\times B$ a total of $6\times 6=36$ independent components. The same argument applies to $R_{iklm}$.

 

[MatinSAR]

Consider the rank-4 product $P_{iklm}=A_{ik}B_{lm}$ of two arbitrary rank-2 tensors $A_{ik}$ and $B_{lm}$. Each tensor $A$ and $B$ has in general $16$ components, so the product $P$ starts with $16\times 16=256$ distinct components. Now impose the two independent antisymmetries:$$P_{iklm}=-P_{kilm}\Rightarrow A_{ik}B_{lm}=-A_{ki}B_{lm}\Rightarrow A\text{ has 6 components}$$$$P_{iklm}=-P_{ikml}\Rightarrow A_{ik}B_{lm}=-A_{ik}B_{ml}\Rightarrow B\text{ has 6 components}$$So the two antisymmetries together give $P=A\times B$ a total of $6\times 6=36$ independent components. The same argument applies to $R_{iklm}$.

This method is different from what my proffessor told us ...
And it is easier. Thank you.

Problem is solved. Thank You, Everyone!

 

[MatinSAR]

Consider the rank-4 product $P_{iklm}=A_{ik}B_{lm}$ of two arbitrary rank-2 tensors $A_{ik}$ and $B_{lm}$. Each tensor $A$ and $B$ has in general $16$ components, so the product $P$ starts with $16\times 16=256$ distinct components. Now impose the two independent antisymmetries:$$P_{iklm}=-P_{kilm}\Rightarrow A_{ik}B_{lm}=-A_{ki}B_{lm}\Rightarrow A\text{ has 6 components}$$$$P_{iklm}=-P_{ikml}\Rightarrow A_{ik}B_{lm}=-A_{ik}B_{ml}\Rightarrow B\text{ has 6 components}$$So the two antisymmetries together give $P=A\times B$ a total of $6\times 6=36$ independent components. The same argument applies to $R_{iklm}$.

Hello. I've thought about your suggested method. What I've understand is that because of $P_{iklm}=-P_{kilm}\Rightarrow A_{ik}B_{lm}=-A_{ki}B_{lm}$ A can only have 6 independent components but components of B are unaffected. So till now we have 16*6=96 independent components.
After applying $P_{iklm}=-P_{ikml}\Rightarrow A_{ik}B_{lm}=-A_{ik}B_{ml}$ Which does not affect A But affects B, B can have only 6 independent components. Applying both gives us a 6*6 matrix with 36 independet components. Am I right?!