268 Antisymmetry
A property of a mapping under which swapping two inputs multiplies the output value by minus one that is used to build determinants of matrices.
definition [d] (Antisymmetry) A property of a tensor or matrix where its components change sign when their indices are interchanged, satisfying the following conditions:
- (Tensor Component Form) For a second-rank tensor \(A\), it is antisymmetric if \(A_{mn} = -A_{nm}\) for all \(m\) and \(n\).
- (Diagonal Property) For any antisymmetric matrix, the diagonal elements (where \(m=n\)) must be zero.
- (Dual Association) In 3-D space, every antisymmetric second-rank tensor \(C\) can be associated with a dual pseudovector \(\mathbf{C}\).
where
- \(A\) is a tensor
- \(m, n\) are indices identifying components
- \(C\) is an antisymmetric tensor