Symmetric Tensor

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

$\begin{displaymath} A^{mn} = A^{nm}. \end{displaymath}$

(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:

$\begin{displaymath} T_{(a,b)} \equiv {\textstyle{1\over 2}}(T_{ab}+T_{ba}) \end{displaymath}$

(3)

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

(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

$\begin{displaymath} a^{11}=a^{22}=0 \end{displaymath}$

(5)

$\begin{displaymath} a^{21}=-a^{12}. \end{displaymath}$

(6)

Let $b_{\alpha\beta}$ be symmetric, so

$\begin{displaymath} b_{12}=b_{21}. \end{displaymath}$

(7)

Then

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