On the Coefficients of Hurwitz-Type Matrix Polynomials
Abstract
Consider the matrix polynomial fn(z)=Iqzn+A1z n−1+⋯+An, where Aj∈ℂq× q. Write fn(z)=hn(z2)+z gn(z2). A matrix polynomial fn is called a Hurwitz-type matrix polynomial if, for n=2m, gn(z)hn(z)−1, and for n=2m+1, hn(z)(zgn(z))−1, admit finite continued fraction expansions with positive definite coefficients. We derive explicit formulas for the coefficients of Hurwitz-type matrix polynomials in terms of orthogonal matrix polynomials, Markov parameters, and Schur complements. We introduce the associated block Hurwitz matrix and establish determinant identities relating it to the corresponding block Hankel matrices. Finally, we settle a conjecture concerning the positivity of the determinants of the coefficient matrices of Hurwitz-type matrix polynomials. We prove that this property holds for degrees at most three, but fails in degree four by means of an explicit counterexample.
Related
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BibTeX
@misc{choquerivero2026coefficients,
title = {{On the Coefficients of Hurwitz-Type Matrix Polynomials}},
author = {Abdon E. Choque-Rivero},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.30089},
eprint = {2608.30089},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30089},
}
RIS
TY - UNPB
TI - On the Coefficients of Hurwitz-Type Matrix Polynomials
AU - Abdon E. Choque-Rivero
PY - 2026
DA - 2026/08/30
PB - arXiv
DO - 10.48550/arxiv.2608.30089
UR - https://arxiv.org/abs/2608.30089
AB - Consider the matrix polynomial f_n(z)=I_qz^n+A_1z^ n−1+⋯+A_n, where A_j∈ℂ^q× q. Write f_n(z)=h_n(z^2)+z g_n(z^2). A matrix polynomial f_n is called a Hurwitz-type matrix polynomial if, for n=2m, g_n(z)h_n(z)^−1, and for n=2m+1, h_n(z)(zg_n(z))^−1, admit finite continued fraction expansions with positive definite coefficients. We derive explicit formulas for the coefficients of Hurwitz-type matrix polynomials in terms of orthogonal matrix polynomials, Markov parameters, and Schur complements. We introduce the associated block Hurwitz matrix and establish determinant identities relating it to the corresponding block Hankel matrices. Finally, we settle a conjecture concerning the positivity of the determinants of the coefficient matrices of Hurwitz-type matrix polynomials. We prove that this property holds for degrees at most three, but fails in degree four by means of an explicit counterexample.
ER -