Relations Between 3- and 4- Vectors and 3 x 3 and 4 x 4 Matrices

Abstract

In one of his last manuscripts Ernst Lüneburg stared at a systematic investigation between 3- and 4- vectors and the various matrix representations occurring in optical and radar polarimetry motivated by Prof. Mott´s exposition about the characteristic features and differences between Kronecker matrices, Kennaugh matrices and 3 x 3 and 4 x 4 covariance matrices. Although the development of this important transformation appears to have taken a long time to be accepted, first results described in this contribution by Ernst Lüneburg are rather promising. The considerations start with the Sinclair or Jones matrix of a coherent target in the linear (x, y) polarization basis and are focusing on the normalized voltage and power equations, knowing well that the optical brightness or radar voltage equations, respectively, are considered to be the cornerstone equations in optical and radar polarimetry. From the preliminary conclusions it appears that the covariance matrix formulation (as 4 x 4 and 3 x 3 matrix) is unable to produce any assessment about optimal power reception even in the coherent scatter case. This is quite in contrast to the Mueller/Kennaugh matrix formulation that involves the same second order matrix elements, but enables one to derive the optimal power reception polarization states.