Antisymmetric second-rank tensor

A second-rank Cartesian tensor is antisymmetric when its components form a skew-symmetric matrix, $T_{ij}=-T_{ji}$. Orthogonal coordinate changes preserve this property because $T^{\prime }=RT{R}^T$ implies $(T^{\prime }{)}^T=R{T}^T{R}^T=-T^{\prime }$.

In three-dimensional Euclidean space, contracting an antisymmetric second-rank tensor with the Levi-Civita symbol gives an axial vector. The epsilon contraction identity proves the inverse formula: $\frac{1}{2}{ϵ}_{kij}{ϵ}_{klm}{A}_{lm}=\frac{1}{2}(A_{ij}-A_{ji})=A_{ij}$. With this index convention,Thus $A$ is minus the cross-product matrix for $a\times x$. Proper rotations transform $a$ as a vector; an orientation-reversing orthogonal change adds a determinant sign. Stating the index convention prevents a sign ambiguity when converting an antisymmetric tensor into a cross-product operator.