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

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##### Journal Title

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##### Volume Title

A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä

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##### Date

2023-09-01

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##### Mcode

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##### Language

en

##### Pages

27

1-27

1-27

##### Series

Linear Algebra and Its Applications, Volume 672

##### Abstract

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.##### Description

Funding Information: Giovanni Barbarino is supported by the Alfred Kordelinin Säätiö Grant No. 210122 . Vanni Noferini is supported by an Academy of Finland grant (Suomen Akatemian päätos 331240 ). Publisher Copyright: © 2023 The Author(s)

##### Keywords

Analytic eigendecomposition, Palindromic matrix polynomial, Para-Hermitian, Para-unitary, Rellich's theorem, Sign characteristic

##### Other note

##### Citation

Barbarino , G & Noferini , V 2023 , ' On the Rellich eigendecomposition of para-Hermitian matrices and the sign characteristics of ⁎-palindromic matrix polynomials ' , Linear Algebra and Its Applications , vol. 672 , pp. 1-27 . https://doi.org/10.1016/j.laa.2023.04.022