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