Prev Subsection A.1.12: Transformation Laws for Cartesian Components of Vectors and Tensors Up Section A.1: Second-Order Tensors Subsection A.1.14: Eigenvalues and Eigenvectors of Real Symmetric Tensors Next
A.1.13 Symmetric and Antisymmetric Tensors
A symmetric tensor satisfies , i.e., , or in matrix form, An antisymmetric (or skew-symmetric) tensor satisfies , i.e., and thus , Any tensor can be written as the sum of a symmetric and antisymmetric tensor,
This is a unique decomposition. It can be checked that is symmetric and is antisymmetric.
The dual vector of an antisymmetric tensor satisfies for any vector . Thus or In matrix form, Conversely, it can also be shown that As a homework problem, it may be shown that , since for any symmetric tensor .