Message Boards Message Boards

0
|
7337 Views
|
2 Replies
|
5 Total Likes
View groups...
Share
Share this post:

Strange results when combining TensorProduct withTensorWedge

Posted 11 years ago
Hello,
I'm trying to figure out the symbolic tensors in Mathematica, but I'm getting some odd results. I have 4 vectors (a, b, c, d)
$Assumptions =
  (a | b | c | d) \[Element] Arrays[{3}, Reals];
and  I want to calculate a tensor product
(a\[TensorWedge]b)\[TensorProduct](c\[TensorWedge]d)
The output of this is
1/4 (a\[TensorProduct]b -
    TensorTranspose[
     a\[TensorProduct]b, {2, 1}])\[TensorProduct](c\[TensorProduct]d -
     TensorTranspose[c\[TensorProduct]d, {2, 1}])
My question is why is that factor of 1/4 there? According to the manual when TensorWedge[a,b] is equal to Multinomial[TensorRank, TensorRank]*
Symmetrize[TensorProduct[a,b], Antisymmetric]. If I substitute this expression in the tensor product, i.e. calculate
(Multinomial[TensorRank[a], TensorRank[b]]*
   Symmetrize[a\[TensorProduct]b,
    Antisymmetric[All]])\[TensorProduct](Multinomial[TensorRank[c],
    TensorRank[d]]*Symmetrize[c\[TensorProduct]d, Antisymmetric[All]])
I get
(a\[TensorProduct]b -
   TensorTranspose[
    a\[TensorProduct]b, {2, 1}])\[TensorProduct](c\[TensorProduct]d -
   TensorTranspose[c\[TensorProduct]d, {2, 1}])
Which is the same as before, but without the 1/4 coefficient. However, if I use numerical tensors and calculate explicitly TensorWedge[a,b] and TensorWedge[b,c] in separate cells and paste the results into
(a\[TensorWedge]b)\[TensorProduct](c\[TensorWedge]d)
I also get the result without the 1/4 coefficient. It seems to me that Mathematica forgets about the Multinomial coefficients when doing the tensor product with symbolic tensors. Does anyone know why this is happening? I haven't found anything in the manual that would explain this behaviour. It seems like a bug to me.

Greetings,
Richard
POSTED BY: Richard Hatz
2 Replies
You can use Trace to follow its logic. The 1/4th value comes from the tranformation of this:
a\[TensorWedge]b
into this
(1/2 (a\[TensorProduct]b - TensorTranspose[a\[TensorProduct]b, {2, 1}]))
You're right that the 1/2 doesn't seem right . Consider the comparison:
{a, b, c}\[TensorWedge]{x, y, z} // Normal
(1/2 ({a, b, c}\[TensorProduct]{x, y, z} - TensorTranspose[{a, b, c}\[TensorProduct]{x, y, z}, {2, 1}]))
POSTED BY: Sean Clarke
Yes, there is a missing multinomial factor in the purely symbolic computation. An unfortunate result of exploring the various possible conventions.

Thank you for reporting this issue!

Jose.
Reply to this discussion
Community posts can be styled and formatted using the Markdown syntax.
Reply Preview
Attachments
Remove
or Discard

Group Abstract Group Abstract