On the Hurwitz Stability of Hurwitz-Type Matrix Polynomials
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912942890418176 |
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| author | Choque-Rivero, Abdon E. |
| author_facet | Choque-Rivero, Abdon E. |
| contents | Every matrix polynomial $\mathbf{f}_n$ can be written in the form \[ \mathbf{f}_n(z)=\mathbf{h}(z^2)+z\,\mathbf{g}_n(z^2). \]
The matrix polynomial $\mathbf{f}_{2m}$ is said to be of Hurwitz type if the expression $\mathbf{g}_{2m}(z)\mathbf{h}_{2m}^{-1}(z)$ admits a representation as a finite continued fraction with positive definite matrix coefficients. Similarly, the odd-degree matrix polynomial $\mathbf{f}_{2m+1}$ is of Hurwitz type if $\frac{1}{z}\mathbf{h}_{2m+1}(z)\mathbf{g}_{2m+1}^{-1}(z)$ has the same property.
We derive an explicit form of the Bezoutian associated with Hurwitz-type matrix polynomials. Using this explicit form, we provide an explicit proof that Hurwitz-type matrix polynomials are Hurwitz matrix polynomials. In [52], the Hurwitzness of Hurwitz-type matrix polynomials was also studied.
Finally, we propose an extension of the class of Hurwitz-type matrix polynomials by adding to a non--Hurwitz-type matrix polynomial another matrix polynomial so that the resulting matrix polynomial is of Hurwitz type. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_10987 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Hurwitz Stability of Hurwitz-Type Matrix Polynomials Choque-Rivero, Abdon E. Classical Analysis and ODEs 34D20, 33C45, 47A56, 15A24, 30E05 Every matrix polynomial $\mathbf{f}_n$ can be written in the form \[ \mathbf{f}_n(z)=\mathbf{h}(z^2)+z\,\mathbf{g}_n(z^2). \] The matrix polynomial $\mathbf{f}_{2m}$ is said to be of Hurwitz type if the expression $\mathbf{g}_{2m}(z)\mathbf{h}_{2m}^{-1}(z)$ admits a representation as a finite continued fraction with positive definite matrix coefficients. Similarly, the odd-degree matrix polynomial $\mathbf{f}_{2m+1}$ is of Hurwitz type if $\frac{1}{z}\mathbf{h}_{2m+1}(z)\mathbf{g}_{2m+1}^{-1}(z)$ has the same property. We derive an explicit form of the Bezoutian associated with Hurwitz-type matrix polynomials. Using this explicit form, we provide an explicit proof that Hurwitz-type matrix polynomials are Hurwitz matrix polynomials. In [52], the Hurwitzness of Hurwitz-type matrix polynomials was also studied. Finally, we propose an extension of the class of Hurwitz-type matrix polynomials by adding to a non--Hurwitz-type matrix polynomial another matrix polynomial so that the resulting matrix polynomial is of Hurwitz type. |
| title | On the Hurwitz Stability of Hurwitz-Type Matrix Polynomials |
| topic | Classical Analysis and ODEs 34D20, 33C45, 47A56, 15A24, 30E05 |
| url | https://arxiv.org/abs/2507.10987 |