Symmetrizing and skew symmetrizing tensors

[quasar_4]

I understand that all rank 2 tensors can be decomposed into a symmetric and a skew symmetric part, but I don't really understood how this is done. It has something to do with permutations of the indices, I guess, but I never learned anything about what a permutation is. Can anyone explain how one would go about symmetrizing (without assuming I have any knowledge of how to permute something)?

 

[robphy]

Just do it...
$$A_{ab}=\frac{A_{ab}+A_{ba}}{2}+\frac{A_{ab}-A_{ba}}{2}$$

To elaborate, for any pair of vectors $$u^a$$ and $$v^b$$
$$A_{ab}u^av^b=\frac{A_{ab}+A_{ba}}{2}u^av^b+\frac{A_{ab}-A_{ba}}{2}u^av^b$$

 

[tacman]

Symmetrizing and skew symmetrizing tensors involves rearranging the indices of the tensor in a specific way to create two new tensors - a symmetric tensor and a skew symmetric tensor. This process is also known as tensor decomposition.

To understand this process, let's first define what a permutation is. A permutation is simply a rearrangement of a set of objects. In the case of tensors, the objects are the indices. For example, if we have a tensor T with indices i, j, and k, a permutation of these indices could be j, i, k. This means that the values of the tensor at position i and j have been swapped.

Now, to symmetrize a tensor, we need to take the original tensor and add it to itself after permuting the indices. This means that for a tensor T with indices i and j, the symmetric tensor S is given by S = 1/2(T + T'), where T' is the original tensor with the indices swapped (i.e. T' = T(j, i)). This process is repeated for all possible permutations of the indices.

On the other hand, to skew symmetrize a tensor, we need to take the original tensor and subtract it from itself after permuting the indices. This means that for a tensor T with indices i and j, the skew symmetric tensor A is given by A = 1/2(T - T'), where T' is the original tensor with the indices swapped (i.e. T' = T(j, i)). This process is also repeated for all possible permutations of the indices.

So, in summary, symmetrizing and skew symmetrizing tensors involves permuting the indices and then adding or subtracting the original tensor from itself to create two new tensors - a symmetric and a skew symmetric tensor. This process is important in tensor analysis as it allows us to break down complex tensors into simpler components for easier analysis and manipulation.