# Sum of tensors

1. Jun 2, 2005

### JohanL

show that $$B_{ij}$$ can be written as the sum of a symmetric tensor
$$B^S_{ij}$$ and an antisymmetric tensor $$B^A_{ij}$$

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

and for an antisymmetric tensor we have
$$B^A_{ij} = -B^A_{ji}$$

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

2. Jun 2, 2005

### robphy

Hint:
define the "symmetric part of B" to be
$$(B_{ij})^S = \frac{1}{2}\left(B_{ij}+ B_{ji}\right)$$
... 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.