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 1.

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.

1. Existence.  Put

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

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

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

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

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

By taking the transposes

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

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

$U^{ij}-U^{\prime ij}$$=$$W^{\prime ij}-W^{ij}$

$=U^{ji}-U^{\prime ji}$$=$$W^{\prime ji}-W^{ji}$

$=W^{ij}-W^{\prime ij}$$=$$U^{\prime ij}-U^{ij}$

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

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