On the Rellich eigendecomposition of para-Hermitian matrices and the sign characteristics of ?-palindromic matrix polynomials

We study the eigendecompositions of para-Hermitian matrices H(z), that is, matrix-valued functions are analytic and Hermitian on unit circle S1?C. In particular, we fill existing gaps in literature prove existence a decomposition H(z)=U(z)D(z)U(z)P where, for all z?S1, U(z) is unitary, U(z)P=U(z)? its conjugate transpose, D(z) real diagonal; moreover, w=z1/N some positive integer N, U(z)P so-called U(z). This generalizes celebrated theorem Rellich line. also show there exists H(z)=V(z)C(z)V(z)P where C(z) pseudo-circulant, V(z) unitary both z. argue that, fact, version Rellich's can be stated function any line or complex plane. Moreover, extend these results to whose entries Puiseux series (that they w but possibly not z). Finally, discuss implications our singular value matrix S1-analytic w, sign characteristics associated with unimodular eigenvalues ?-palindromic polynomials.