Resistivity tensor ($E_i={\sum }_j{\rho }_{ij}{J}_j$)

The resistivity tensor is the matrix relating electric field to current density in linear local transport. It is the matrix inverse of the conductivity tensor, not the componentwise reciprocal. A Hall response produces antisymmetric off-diagonal components. In a multicarrier system, the conductivities add because each species sees the same electric field; invert only after summing the species' conductivity tensors.