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Para-Hermitian rational matrices

Froilán Dopico, Vanni Noferini, María C. Quintana, Paul Van Dooren

TL;DR

The paper addresses structure-preserving linearization of para-Hermitian and para-skew-Hermitian rational matrices by constructing strongly minimal $*$-palindromic linearizations for $(1+z)R(z)$ using Möbius transforms that map unit-circle Hermitian structure to real-line Hermitian structure. It proves an impossibility result for direct $*$-palindromic linearizations of $R(z)$ and develops a constructive framework based on a stable/anti-stable decomposition, along with Taylor and partial fraction representations of the Rin part, to obtain explicit linearizations that preserve pole-zero symmetries and minimal indices. It provides parameterizations, alternative Möbius approaches (including a family $B_\alpha$), and methods to handle unit-circle poles by combining in/out/p-P components, with detailed procedures for building linearizations from Rin’s representations. The work advances practical structure-preserving eigenvalue computations in applications like spectral factorization and control, while leaving numerical stability analysis and further refinements for future study.

Abstract

In this paper we study para-Hermitian rational matrices and the associated structured rational eigenvalue problem (REP). Para-Hermitian rational matrices are square rational matrices that are Hermitian for all $z$ on the unit circle that are not poles. REPs are often solved via linearization, that is, using matrix pencils associated to the corresponding rational matrix that preserve the spectral structure. Yet, non-constant polynomial matrices cannot be para-Hermitian. Therefore, given a para-Hermitian rational matrix $R(z)$, we instead construct a $*$-palindromic linearization for $(1+z)R(z)$, whose eigenvalues that are not on the unit circle preserve the symmetries of the zeros and poles of $R(z)$. This task is achieved via Möbius transformations. We also give a constructive method that is based on an additive decomposition into the stable and anti-stable parts of $R(z)$. Analogous results are presented for para-skew-Hermitian rational matrices, i.e., rational matrices that are skew-Hermitian upon evaluation on those points of the unit circle that are not poles.

Para-Hermitian rational matrices

TL;DR

The paper addresses structure-preserving linearization of para-Hermitian and para-skew-Hermitian rational matrices by constructing strongly minimal -palindromic linearizations for using Möbius transforms that map unit-circle Hermitian structure to real-line Hermitian structure. It proves an impossibility result for direct -palindromic linearizations of and develops a constructive framework based on a stable/anti-stable decomposition, along with Taylor and partial fraction representations of the Rin part, to obtain explicit linearizations that preserve pole-zero symmetries and minimal indices. It provides parameterizations, alternative Möbius approaches (including a family ), and methods to handle unit-circle poles by combining in/out/p-P components, with detailed procedures for building linearizations from Rin’s representations. The work advances practical structure-preserving eigenvalue computations in applications like spectral factorization and control, while leaving numerical stability analysis and further refinements for future study.

Abstract

In this paper we study para-Hermitian rational matrices and the associated structured rational eigenvalue problem (REP). Para-Hermitian rational matrices are square rational matrices that are Hermitian for all on the unit circle that are not poles. REPs are often solved via linearization, that is, using matrix pencils associated to the corresponding rational matrix that preserve the spectral structure. Yet, non-constant polynomial matrices cannot be para-Hermitian. Therefore, given a para-Hermitian rational matrix , we instead construct a -palindromic linearization for , whose eigenvalues that are not on the unit circle preserve the symmetries of the zeros and poles of . This task is achieved via Möbius transformations. We also give a constructive method that is based on an additive decomposition into the stable and anti-stable parts of . Analogous results are presented for para-skew-Hermitian rational matrices, i.e., rational matrices that are skew-Hermitian upon evaluation on those points of the unit circle that are not poles.
Paper Structure (11 sections, 16 theorems, 77 equations, 1 table)

This paper contains 11 sections, 16 theorems, 77 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. It is impossible to linearize a para-Hermitian rational matrix with a palindromic system matrix
  4. Para-Hermitian rational matrices and Möbius transform
  5. Alternative approaches: The choice of the Möbius transform
  6. Parametrizing para-Hermitian rational matrices
  7. Decomposition into stable and anti-stable parts
  8. Construction of linearizations from different representations of $R_{in}(z)$
  9. Construction from the Taylor expansion of $R_{in}(z)$
  10. Construction from the partial fraction decomposition of $R_{in}(z)$
  11. Conclusions and future work

Key Result

Theorem 3.1

\newlabelthm:$*$-palindromic_lsm0 Consider a $*$-palindromic (resp., $*$-anti-palindromic) linear system matrix $L(z)$. Then the following statements hold:

Theorems & Definitions (29)

  • Definition 1.1
  • Theorem 3.1
  • Proof 1
  • Corollary 3.2
  • Proof 2
  • Lemma 4.1
  • Proof 3
  • Theorem 4.3
  • Proof 4
  • Proof 5
  • ...and 19 more