In metal detection, the hidden detected metallic object can be represented by its first order polarization tensor. The polarization tensor contains significant information about the hidden object such as the size and the material of the object, which could be helpful in identifying the object before any further actions are taken regarding the object. Moreover, given the polarization tensor associated with the hidden object, an ellipsoid that has the same order polarization tensor can also be determined and described, where the physical properties of the ellipsoid might be similar to the physical of the actual hidden object. Therefore, in this study, a property of the ellipsoid is investigated, such that, the orientation of an ellipsoid is identified based on a given polarization tensor that represents the ellipsoid. Specifically, given a first order polarization tensor representing an ellipsoid after it is being rotated, it is possible to determine its eigenvalues and eigenvectors that correspond to the eigenvalues. Next, the nature of the elements in first order polarization tensor is investigated to be either a positive or negative definite matrix by using its eigenvalues. After that, the rotated first order polarization is further justified to be diagonalizable under certain conditions. The orientation of the ellipsoid after it is rotated can then be described by the related axis-aligned ellipsoid together with the angle of rotation applied.