Para-Hermitian Rational Matrices

Froilán M. Dopico, Vanni Noferini, Maria C. Quintana, Paul van Dooren

Research output: Contribution to journalArticleScientificpeer-review

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, nonconstant 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 antistable 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.

Original languageEnglish
Pages (from-to)2339-2359
Number of pages21
JournalSIAM Journal on Matrix Analysis and Applications
Volume45
Issue number4
DOIs
Publication statusPublished - 2024
MoE publication typeA1 Journal article-refereed

Keywords

  • *-palindromic
  • linear System matrices
  • linearization
  • Mobius
  • para-Hermitian
  • rational matrices
  • strong minimality

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