🧄Tensor Symmetrization and Anti-Symmetrization Properties

Any tensor can be uniquely broken down into a symmetric part and an anti-symmetric part. A tensor is symmetric if it remains unchanged when its indices are swapped ( $S_{i j}=S_{j i}$ ), while it's anti-symmetric if swapping indices changes its sign $\left(A_{i j}=-A_{j i}\right)$. The core principle is that if a tensor is "purely" one type, the other part vanishes. For example, the symmetric component of an anti-symmetric tensor is zero, as the definition of anti-symmetry ( $A_{i j}=-A_{j i}$ ) cancels out the terms in the symmetrization formula. Similarly, the anti-symmetric component of a symmetric tensor is zero because the terms in the anti-symmetrization formula cancel out. Conversely, applying a symmetrization operation to a symmetric tensor, or an anti-symmetrization operation to an anti-symmetric tensor, simply returns the original tensor, as they are already in their "pure" form. This confirms that these operations are designed to isolate a specific property of the tensor without altering its fundamental nature.

Tensor Symmetrization and Anti-Symmetrization Properties | Proof and Derivationviadean on Notion