According to previous studies by Ammari (2007) and Nelson (2009), electro-sensing by a weakly electric fish has some similarities with the electrical impedance tomography (EIT) system. Because of this we adapted the polarization tensor from EIT into the electro-sensing context. Our aim is to investigate object characterization by the fish through the first order polarization tensor without actually reconstructing an image of the object. In this talk, I will discuss the mathematical background of the polarization tensor in electro-sensing. After that, some examples of object characterization through the tensor by the fish will be given.