Para-Hermitian rational matrices

TL;DR

This paper investigates para-Hermitian rational matrices and develops methods to linearize associated eigenvalue problems while preserving spectral symmetries, using M"{o}bius transformations and matrix decompositions.

Contribution

It introduces a novel linearization approach for para-Hermitian rational matrices that maintains spectral symmetry through $*$-palindromic linearizations and M"{o}bius transformations.

Findings

Constructed $*$-palindromic linearizations for para-Hermitian matrices.

Provided a decomposition method into stable and anti-stable parts.

Extended results to para-skew-Hermitian matrices.

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 M\"{o}bius 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.