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second order tensor: symmetric and skew-symmetric parts
We shall prove the following theorem on existence and uniqueness. (Here, we assime that the ground field has characteristic different from 2. This hypothesis is satisfied for the cases of greatest interest, namely real and complex ground fields.)
Theorem Every covariant and contravariant tensor of second rank may be expressed univocally as the sum of a symmetric and skew-symmetric tensor.
Proof. Let us consider a contravariant tensor.
Then $U^{ij}=U^{ji}$ is symmetric, $W^{ij}=-W^{ji}$ is skew-symmetric, and
2. Uniqueness. Let us suppose that $T^{ij}$ admits the decompositions
By taking the transposes
we separate the symmetric and skew-symmetric parts in both equations and making use of their symmetry properties, we have \begin{eqnarray*} U^{ij}-U'^{ij} & = & W'^{ij}-W^{ij} \\ =U^{ji}-U'^{ji} & = & W'^{ji}-W^{ji} \\ =W^{ij}-W'^{ij} & = & U'^{ij}-U^{ij} \\ =-(U^{ij}-U'^{ij}) & = & 0, \end{eqnarray*}which shows uniqueness of each part. mutatis mutandis for a covariant tensor $T_{ij}$ .
1. Existence. Put
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Then $U^{ij}=U^{ji}$ is symmetric, $W^{ij}=-W^{ji}$ is skew-symmetric, and
 |
2. Uniqueness. Let us suppose that $T^{ij}$ admits the decompositions
 |
By taking the transposes
 |
we separate the symmetric and skew-symmetric parts in both equations and making use of their symmetry properties, we have \begin{eqnarray*} U^{ij}-U'^{ij} & = & W'^{ij}-W^{ij} \\ =U^{ji}-U'^{ji} & = & W'^{ji}-W^{ji} \\ =W^{ij}-W'^{ij} & = & U'^{ij}-U^{ij} \\ =-(U^{ij}-U'^{ij}) & = & 0, \end{eqnarray*}which shows uniqueness of each part. mutatis mutandis for a covariant tensor $T_{ij}$ .
second order tensor: symmetric and skew-symmetric parts is owned by Raymond Puzio, Pedro A. Fernandez, Thomas Foregger.
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