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Because the polarization sense of an electromagnetic wave is changed when the direction of propagation is reversed, a new type of polarization vector which is directionally dependent is introduced. The scattering matrix formulation is then introduced in terms of directional vectors and directional transformation matrices, and the transformation of the scattering matrix under a unitary change of polarization basis transformation is shown to be a congruent transformation. The congruent sub-group of unitary transformations of the polarization basis is then discussed, and it is shown that the scattering matrix can be reduced to diagonal form by this sub-group of transformations. A new matrix, called the polarization power scattering matrix, is then introduced and its relation to the scattering matrix is discussed. The power matrix gives the total power back-scattered from the target for any transmitted polarization. It specifies the scattering matrix up to two phase angles (one of which is of no importance), and is more easily measured. The total power scattering matrix can be determined for any target by measuring only the total power in the backscattered return; no phase measurement is necessary and only plane polarizations need be transmitted.