# Tensor Symmetrization and Anti-Symmetrization Properties

- Decomposition: Any tensor can be uniquely decomposed into a symmetric part and an anti-symmetric part.
- Definitions:
- A symmetric tensor remains unchanged when its indices are swapped ( $S_{i j}=S_{j i}$ ).
- An anti-symmetric tensor changes its sign when its indices are swapped ( $A_{i j}=-A_{j i}$ ).
- Vanishing Components: The symmetric part of a purely anti-symmetric tensor is zero, and the anti-symmetric part of a purely symmetric tensor is zero. This confirms that these tensors are "pure" and lack the opposite component.
- Self-Identity: The symmetric part of a symmetric tensor is the tensor itself, and the antisymmetric part of an anti-symmetric tensor is the tensor itself. The operations of symmetrization or anti-symmetrization don't change a tensor that already possesses that specific property.

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