Sum of tensors

  • Thread starter JohanL
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show that [tex]B_{ij}[/tex] can be written as the sum of a symmetric tensor
[tex]B^S_{ij}[/tex] and an antisymmetric tensor [tex]B^A_{ij}[/tex]

i dont know how to do this one.
for a symmetric tensor we have
[tex]B^S_{ij} = B^S_{ji}[/tex]

and for an antisymmetric tensor we have
[tex]B^A_{ij} = -B^A_{ji}[/tex]

the only thing my book says is that the sum should be a tensor of the same type.
 

robphy

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Hint:
define the "symmetric part of B" to be
[tex](B_{ij})^S = \frac{1}{2}\left(B_{ij}+ B_{ji}\right)[/tex]
... quite analogous to defining the "real part of a complex number z" as (z+z*)/2. [Check for yourself that this "symmetric part" is truly symmetric.]

I'm sure you can finish the rest.
 

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