# second order tensor: symmetric and skew-symmetric parts

###### Proof.

Let us consider a contravariant tensor.

1. Existence.  Put

 $\displaystyle U^{ij}=\frac{1}{2}(T^{ij}+T^{ji}),\qquad W^{ij}=\frac{1}{2}(T^{% ij}-T^{ji}).$

Then $U^{ij}=U^{ji}$ is symmetric, $W^{ij}=-W^{ji}$ is skew-symmetric, and

 $\displaystyle T^{ij}=U^{ij}+W^{ij}.$

2. Uniqueness.  Let us suppose that $T^{ij}$ admits the decompositions

 $\displaystyle T^{ij}=U^{ij}+W^{ij}=U^{\prime ij}+W^{\prime ij}.$
 $\displaystyle T^{ji}=U^{ji}+W^{ji}=U^{\prime ji}+W^{\prime ji},$

we separate the symmetric and skew-symmetric parts in both equations and making use of their symmetry properties, we have

 $\displaystyle U^{ij}-U^{\prime ij}$ $\displaystyle=$ $\displaystyle W^{\prime ij}-W^{ij}$ $\displaystyle=U^{ji}-U^{\prime ji}$ $\displaystyle=$ $\displaystyle W^{\prime ji}-W^{ji}$ $\displaystyle=W^{ij}-W^{\prime ij}$ $\displaystyle=$ $\displaystyle U^{\prime ij}-U^{ij}$ $\displaystyle=-(U^{ij}-U^{\prime ij})$ $\displaystyle=$ $\displaystyle 0,$

which shows uniqueness of each part. mutatis mutandis  for a covariant tensor $T_{ij}$. ∎

Title second order  tensor: symmetric and skew-symmetric parts SecondOrderTensorSymmetricAndSkewsymmetricParts 2013-03-22 15:51:32 2013-03-22 15:51:32 rspuzio (6075) rspuzio (6075) 18 rspuzio (6075) Theorem msc 15A69