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Abstract
We study the eigendecompositions of paraHermitian matrices H(z), that is, matrixvalued functions that are analytic and Hermitian on the unit circle S^{1}⊂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∈S^{1}, 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=z^{1/N} for some positive integer N, and U(z)^{P} is the socalled paraHermitian conjugate of U(z). This generalizes the celebrated theorem of Rellich for matrixvalued 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 pseudocirculant, 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 matrixvalued function that are analytic and Hermitian on any line or any circle on the complex plane. Moreover, we extend these results to paraHermitian 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 S^{1}analytic functions of w, and on the sign characteristics associated with unimodular eigenvalues of ⁎palindromic matrix polynomials.
Original language  English 

Pages (fromto)  127 
Number of pages  27 
Journal  Linear Algebra and Its Applications 
Volume  672 
DOIs  
Publication status  Published  1 Sept 2023 
MoE publication type  A1 Journal articlerefereed 
Keywords
 Analytic eigendecomposition
 Palindromic matrix polynomial
 ParaHermitian
 Paraunitary
 Rellich's theorem
 Sign characteristic
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Noferini_Vanni_AoF_Project: Noferini Vanni Academy Project
Noferini, V., Quintana Ponce, M., Barbarino, G., Wood, R. & Nyman, L.
01/09/2020 → 31/08/2024
Project: Academy of Finland: Other research funding