Physlib.Relativity.Tensors.ComplexTensor.Lemmas

Let $A$ be a complex Lorentz tensor with two contravariant indices ($A \in \mathbb{C}T^{\mu\nu}$) and $S$ be a complex Lorentz tensor with two covariant indices ($S \in \mathbb{C}T_{\mu\nu}$). If $A$ is antisymmetric, such that $A^{\mu\nu} = -A^{\nu\mu}$, and $S$ is symmetric, such that $S_{\mu\nu} = S_{\nu\mu}$, then their full contraction satisfies the identity $A^{\mu\nu} S_{\mu\nu} = -A^{\mu\nu} S_{\mu\nu}$.