second order tensor: symmetric and skew-symmetric parts

We shall prove the following theoremMathworldPlanetmath on existence and uniqueness. (Here, we assime that the ground field has characteristic different from 2. This hypothesisMathworldPlanetmathPlanetmath 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 symmetricPlanetmathPlanetmathPlanetmathPlanetmath and skew-symmetric tensor.


Let us consider a contravariant tensor.

1. Existence.  Put


Then Uij=Uji is symmetric, Wij=-Wji is skew-symmetric, and


2. Uniqueness.  Let us suppose that Tij admits the decompositions


By taking the transposesMathworldPlanetmath


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

Uij-Uij = Wij-Wij
=Uji-Uji = Wji-Wji
=Wij-Wij = Uij-Uij
=-(Uij-Uij) = 0,

which shows uniqueness of each part. mutatis mutandis  for a covariant tensor Tij. ∎

Title second orderPlanetmathPlanetmath tensor: symmetric and skew-symmetric parts
Canonical name SecondOrderTensorSymmetricAndSkewsymmetricParts
Date of creation 2013-03-22 15:51:32
Last modified on 2013-03-22 15:51:32
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 18
Author rspuzio (6075)
Entry type Theorem
Classification msc 15A69