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Symmetric Tensor

A second-Rank symmetric Tensor is defined as a Tensor $A$ for which

A^{mn} = A^{nm}.
\end{displaymath} (1)

Any Tensor can be written as a sum of symmetric and Antisymmetric parts
$\displaystyle A^{mn}$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}(A^{mn}+A^{nm})+{\textstyle{1\over 2}}(A^{mn}-A^{nm})$  
  $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}({B_S}^{mn}+{B_A}^{mn}).$ (2)

The symmetric part of a Tensor is denoted by parentheses as follows:
T_{(a,b)} \equiv {\textstyle{1\over 2}}(T_{ab}+T_{ba})
\end{displaymath} (3)

T_{(a_1,a_2,\ldots,a_n)} \equiv {1\over n!} \sum_{\rm permutations} T_{a_1a_2\cdots a_n}.
\end{displaymath} (4)

The product of a symmetric and an Antisymmetric Tensor is 0. This can be seen as follows. Let $a^{\alpha \beta}$ be Antisymmetric, so

\end{displaymath} (5)

\end{displaymath} (6)

Let $b_{\alpha\beta}$ be symmetric, so
\end{displaymath} (7)

$\displaystyle a^{\alpha \beta }b_{\alpha \beta}$ $\textstyle =$ $\displaystyle a^{11}b_{11}+a^{12}b_{12}+a^{21}b_{21}+a^{22}b_{22}$  
  $\textstyle =$ $\displaystyle 0+a^{12}b_{12}-a^{12}b_{12}+0=0.$ (8)

A symmetric second-Rank Tensor $A_{mn}$ has Scalar invariants

$\displaystyle s_1$ $\textstyle =$ $\displaystyle A_{11}+A_{22}+A_{22}$ (9)
$\displaystyle s_2$ $\textstyle =$ $\displaystyle A_{22}A_{33}+A_{33}A_{11}+A_{11}A_{22}-{A_{23}}^2-{A_{31}}^2-{A_{12}}^2.$ (10)

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© 1996-9 Eric W. Weisstein