Work

Para-Hermitian Rational Matrices

Year: 2024

Type: article

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 [Formula: see text] 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 [Formula: see text], we instead construct a [Formula: see text]-palindromic linearization for [Formula: see text], whose eigenvalues that are not on the unit circle preserve the symmetries of the zeros and poles of [Formula: see text]. 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 [Formula: see text]. 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.

Source: SIAM Journal on Matrix Analysis and Applications

DOI: 10.1137/24m1678416

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

Institutions: Universidad Carlos III de Madrid, Aalto University, UCLouvain

Language: English

Topic: Matrix Theory and Algorithms

Subfield: Computational Theory and Mathematics

Field: Computer Science

Domain: Physical Sciences

Open Access status: green

Funder: Agencia Estatal de Investigación