# Is there a mistake in this tensor multiplication problem?

• DuckAmuck
In summary: I am just trying to figure out *how* this could be zero at this point, as in what conditions. Otherwise I’m stumped.
DuckAmuck
Homework Statement
Show that
epsilon_{ijkl} ( M^{ij} N^{kl} + N^{ij} M^{kl}) = 0
Relevant Equations
epsilon is the 4D anti-symmetric Levi-Cevita tensor. M and N are also anti-symmetric tensors.
ep_{ijkl} M^{ij} N^{kl} + ep_{ijkl}N^{ij} M^{kl}
The second term can be rewritten with indices swapped
ep_{klij} N^{kl}M^{ij}
Shuffle indices around in epsilon
ep{klij} = ep{ijkl}
Therefore the expression becomes
2ep_{ijkl}M^{ij}N^{kl}
Not zero.
What is wrong here?

I tried, but got the same result as you did. Are you sure its not supposed to be
##\epsilon_{ijkl} ( M^{ij} N^{kl} - N^{ij} M^{kl}) = 0##?
What I did was to write everything out, using all even permutations of 1,2,3,4:
{1,2,3,4}, {1,3,4,2}, {1,4,2,3}, {2,1,4,3}, {2,3,1,4}, {2,4,3,1}, {3,1,2,4}, {3,2,4,1}, {3,4,1,2}, {4,1,3,2}, {4,2,1,3}, {4,3,2,1}
and all odd ones:
{1,2,4,3}, {1,3,2,4}, {1,4,3,2}, {2,1,3,4}, {3,2,1,4}, {4,2,3,1}, {2,3,4,1}, {2,4,1,3}, {3,1,4,2}, {3,4,2,1}, {4,1,2,3}, {4,3,1,2}
and the fact that ##M## and ##N## are anti-symmetrical, i.e. ##M^{12}= - M^{21}## etc.

ok i think i have solid reasoning here:

Suppose ##C^{ij} = M^{ij} + N^{ij}##

From symmetry and antisymmetry we have:

##\epsilon_{ijkl} C^{ij}C^{kl} = 0##

Also if you foil the CC product in terms of M and N you get ##C^{ij}C^{kl} = M^{ij}M^{kl} + N^{ij}N^{kl} + M^{ij}N^{kl} + N^{ij}M^{kl}##

The MM and NN terms are zero for the same reason the CC product is when multiplied by epsilon.

So this demands that

##\epsilon_{ijkl} (M^{ij}N^{kl} + N^{ij}M^{kl}) = 0##

DuckAmuck said:
ok i think i have solid reasoning here:

Suppose ##C^{ij} = M^{ij} + N^{ij}##

From symmetry and antisymmetry we have:

##\epsilon_{ijkl} C^{ij}C^{kl} = 0##

Also if you foil the CC product in terms of M and N you get ##C^{ij}C^{kl} = M^{ij}M^{kl} + N^{ij}N^{kl} + M^{ij}N^{kl} + N^{ij}M^{kl}##

The MM and NN terms are zero for the same reason the CC product is when multiplied by epsilon.

So this demands that

##\epsilon_{ijkl} (M^{ij}N^{kl} + N^{ij}M^{kl}) = 0##

No, the statement as it stands seems false to me. It is not generally the case thar ##\epsilon_{ijkl} C^{ij} C^{kl} = 0##.

malawi_glenn
Orodruin said:
No, the statement as it stands seems false to me. It is not generally the case thar ##\epsilon_{ijkl} C^{ij} C^{kl} = 0##.
You’re right. I am just trying to figure out *how* this could be zero at this point, as in what conditions. Otherwise I’m stumped.

DuckAmuck said:
I am just trying to figure out *how* this could be zero at this point, as in what conditions.

You should have been given all the conditions already, that M and N are antisymmetric rank-2 tensors.

There is always the possibility that whoever gave you this problem, is wrong / made a typo. I have been tearing my hair off several times doing excersices in general relativity books... to find out there was some typo in the problem as written.

Here is my "expanded" calculation that I did btw:

The underlined terms I will collect at the end.

## \underline{M^{12}N^{34}} + M^{13}N^{42} + M^{14}N^{23} + \underline{M^{21}N^{43}} + M^{23}N^{14} + M^{24}N^{31} + M^{31}N^{24} + M^{32}N^{41} + M^{34}N^{12} + M^{41}N^{32} + M^{42}N^{13} + M^{43}N^{21} ##
##- ( \underline{M^{12}N^{43}} + M^{13}N^{24} + M^{14}N^{32} + \underline{M^{21}N^{34}} + M^{32}N^{14} + M^{42}N^{31} + M^{23}N^{41} + M^{24}N^{13} + M^{31}N^{42} + M^{34}N^{21} + M^{41}N^{23} + M^{43}N^{12} )##
##+ N^{12}M^{34} + N^{13}M^{42} + N^{14}M^{23} + N^{21}M^{43} + N^{23}M^{14} + N^{24}M^{31} + N^{31}M^{24} + N^{32}M^{41} + \underline{N^{34}M^{12}} + N^{41}M^{32} + N^{42}M^{13} + \underline{N^{43}M^{21}} ##
##- ( N^{12}M^{43} + N^{13}M^{24} + N^{14}M^{32} + N^{21}M^{34} + N^{32}M^{14} + N^{42}M^{31} + N^{23}M^{41} + N^{24}M^{13} + N^{31}M^{42} + \underline{N^{34}M^{21}} + N^{41}M^{23} + \underline{N^{43}M^{12}} \: ) ##

The stuff I underlined:
## M^{12}N^{34} + M^{21}N^{43} - M^{12}N^{43} - M^{21}N^{34} + N^{34}M^{12} +N^{43}M^{21} -N^{34}M^{21} - N^{43}M^{12} ##

(##M^{21}= - M^{12}## and ##N^{43}= - N^{34}##)

##M^{12}N^{34} + (-1)^2 M^{12}N^{34} - (-1)M^{12}N^{34} - (-1)M^{12}N^{34} + N^{34}M^{12} +(-1)^2N^{34}M^{12} - (-1)N^{34}M^{12} - (-1)N^{34}M^{12} = 8M^{12}N^{34} ##

Well that was fun.

DuckAmuck

## 1. What is a tensor multiplication problem?

A tensor multiplication problem involves multiplying tensors, which are multidimensional arrays, according to certain rules and conventions. It is a common operation in linear algebra and is used in various fields of science and engineering.

## 2. How is tensor multiplication different from regular matrix multiplication?

Tensor multiplication is different from regular matrix multiplication because it involves multiplying multidimensional arrays, whereas regular matrix multiplication involves multiplying two-dimensional arrays. Additionally, tensor multiplication follows different rules and conventions than regular matrix multiplication.

## 3. What are some common applications of tensor multiplication?

Tensor multiplication is used in various fields of science and engineering, including physics, computer science, and machine learning. It is commonly used in data analysis and image processing, as well as in solving differential equations and modeling physical systems.

## 4. What are some strategies for solving tensor multiplication problems?

There are several strategies for solving tensor multiplication problems, including using index notation, Einstein summation convention, and graphical representations. It is also important to understand the properties of tensors, such as symmetry and invertibility, to effectively solve these problems.

## 5. Are there any tools or software that can help with tensor multiplication?

Yes, there are various tools and software that can assist with tensor multiplication, such as MATLAB, Python's NumPy library, and WolframAlpha. These tools provide functions and algorithms for performing tensor operations and can also help with visualizing and understanding the results of tensor multiplication.

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