New method for general Kennaugh's pseudo-eigenvalue equation in radar polarimetry
This paper studies the Kennaugh pseudo-eigenvalue equation \( Sx = \lambda x^*\) that is basic to radar polarimetry. This equation involves the coneigenvalues of \(S\) and it can be rephrased in a Gaussian normal equations way as \(SS^*x = |\lambda|^2x\). Thus with \(A = SS^*\) it becomes a special case of \[ AX^* - XB = O_n \;. \tag{\(*\)} \] This latter equation has been studied extensively and for varying right hand sides \(C\). The authors replace \(A_{n,n} = A_1 + iA_2\) by its real representation \(A_\sigma = \left( \begin{matrix} A_1 & A_2\\A_2 & -A_1\end{matrix} \right)\) and relate the solvability of (\(*\)) to the matrix equation \(A_\sigma Y - Y B_\sigma = C _\sigma \in C^{2n,2n}\) having a real matrix solution (Theorem 1). This is extended and in turn classifies the coneigenvalues and coneigenvectors of \(A = SS^*\). Finally, a method for finding the coneigenvalues/coneigenvectors in this way is explained.
scientific article; zbMATH DE number 6107739