Tensor

A tensor is a geometric object with indexed components. A tensor can have any number of indices in the up and down positions, and the number of indices determines its rank. For example:

The metric of spacetime $\eta_{\mu\nu}$ is a rank $(0,2)$ tensor.
A matrix $A^\mu{}_\nu$ is a rank $(1,1)$ tensor.
An up vector $x^\alpha$ is rank $(1,0)$ , and a down vector $x_\alpha$ is rank $(0,1)$ . Both are tensors.
A scalar is considered (by some, including me) a rank $(0,0)$ tensor.

A quantity with up and down indices is a tensor if it transforms between reference frames using one Lorentz transform for each index. For example, if we have a tensor $R^\mu{}_{\alpha\beta\gamma}$ , it must transform following

\begin{align*} R^{\mu'}{}_{\alpha'\beta'\gamma'} = \Lambda^{\mu'}{}_\mu \Lambda^\alpha{}_{\alpha'} \Lambda^\beta{}_{\beta'} \Lambda^\gamma{}_{\gamma'} R^\mu{}_{\alpha\beta\gamma}. \end{align*}

Rank-2 tensors sometimes have special properties.

A tensor $S^{\alpha\beta}$ is symmetric if $S^{\alpha\beta} = S^{\beta\alpha}$ . A symmetric matrix has 10 independent components.
A tensor $S^{\alpha\beta}$ is antisymmetric if $S^{\alpha\beta} = -S^{\beta\alpha}$ . In this case the diagonal must be zero and the other components have a flipped sign across the diagonal. An antisymmetric matrix has 6 independent components.