# 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 ${T}_{ij}$. ∎

Title | second order^{} 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 |