Perturbation of monic matrix polynomials
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arXiv
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| Format: | Preprint |
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2026
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| author | Le, Cong Trinh Lee, Gue Myung Lim, Yongdo Pham, Tien Son |
| author_facet | Le, Cong Trinh Lee, Gue Myung Lim, Yongdo Pham, Tien Son |
| contents | In this paper, we study the stability of matrix polynomials under structured perturbations of their coefficients. More precisely, we consider a family of matrix polynomials \[ P_u(λ)=A_d(u)λ^d+A_{d-1}(u)λ^{d-1}+\cdots+A_0(u), \] whose matrix coefficients depend continuously and semialgebraically on a parameter vector $u\in\mathbb{C}^p$.
Assuming that the matrix polynomial is monic, we show that the spectrum, the $\varepsilon$-pseudospectrum, the numerical range, and the joint numerical range associated with $P_u(λ)$ define set-valued maps that are Hölder continuous with respect to the parameter $u$. Moreover, the parameter space $\mathbb{C}^p$ can be decomposed into a finite union of analytic semialgebraic submanifolds such that, on each submanifold, the eigenvalues and the Jordan pairs of $P_u(λ)$ depend analytically on $u$.
We also note that most of the results remain valid if the monicity assumption is replaced by the local nonsingularity of the leading coefficient matrix $A_d(u)$. However, the monic setting is adopted throughout the paper in order to simplify the exposition and to avoid additional technical assumptions, which are required in particular for results concerning numerical ranges. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_00036 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Perturbation of monic matrix polynomials Le, Cong Trinh Lee, Gue Myung Lim, Yongdo Pham, Tien Son Rings and Algebras Numerical Analysis Complex Variables Spectral Theory 15A18, 15A22, 65F15, 65F35, 14P10, 32B20 In this paper, we study the stability of matrix polynomials under structured perturbations of their coefficients. More precisely, we consider a family of matrix polynomials \[ P_u(λ)=A_d(u)λ^d+A_{d-1}(u)λ^{d-1}+\cdots+A_0(u), \] whose matrix coefficients depend continuously and semialgebraically on a parameter vector $u\in\mathbb{C}^p$. Assuming that the matrix polynomial is monic, we show that the spectrum, the $\varepsilon$-pseudospectrum, the numerical range, and the joint numerical range associated with $P_u(λ)$ define set-valued maps that are Hölder continuous with respect to the parameter $u$. Moreover, the parameter space $\mathbb{C}^p$ can be decomposed into a finite union of analytic semialgebraic submanifolds such that, on each submanifold, the eigenvalues and the Jordan pairs of $P_u(λ)$ depend analytically on $u$. We also note that most of the results remain valid if the monicity assumption is replaced by the local nonsingularity of the leading coefficient matrix $A_d(u)$. However, the monic setting is adopted throughout the paper in order to simplify the exposition and to avoid additional technical assumptions, which are required in particular for results concerning numerical ranges. |
| title | Perturbation of monic matrix polynomials |
| topic | Rings and Algebras Numerical Analysis Complex Variables Spectral Theory 15A18, 15A22, 65F15, 65F35, 14P10, 32B20 |
| url | https://arxiv.org/abs/2603.00036 |