2025/07/15 by Abdon E. Choque‐Rivero, Choque-Rivero, Abdon E.
Computer Science · Mathematics · #15A24 #30E05 #33C45 #34D20 #47A56 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Holomorphic and Operator Theory #Matrix Theory and Algorithms #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2507.10987
openalex publication_date 2025/07/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Every matrix polynomial \mathbf fn admits a decomposition of the form \mathbf fn(z)=\mathbf hn(z2)+z \mathbf gn(z2). The matrix polynomial \mathbf f2m is said to be of Hurwitz type if the expression \mathbf g2m(z)\mathbf h2m-1(z) admits a representation as a finite continued fraction with positive definite matrix coefficients. Similarly, the odd-degree matrix polynomial \mathbf f2m+1 is of Hurwitz type if (1)/(z)\mathbf h2m+1(z)\mathbf g2m+1-1(z) has the same property. We derive an explicit representation of the Bezoutian associated with Hurwitz-type matrix polynomials. Using this representation, we obtain a direct proof that every Hurwitz-type matrix polynomial is Hurwitz. The Hurwitz property of this class was also investigated in [52]; our approach is based on an explicit Bezoutian representation. This provides a constructive connection between continued-fraction representations, matrix Bezoutians, and Hurwitz stability. We also develop a completion procedure that associates with a given matrix polynomial a Hurwitz-type matrix polynomial of higher degree. As a consequence, whenever such a completion exists, the original polynomial is Hurwitz. The proposed construction is illustrated by examples.