Structured Linearizations of Structured Rational Matrices
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866911467095195648 |
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| author | Bist, Avisek Behera, Namita |
| author_facet | Bist, Avisek Behera, Namita |
| contents | Numerical computations involving rational matrices often benefit from preserving underlying matrix structures such as symmetry, Hermitian properties, or sparsity that reflect physical, geometric, or algebraic characteristics of the system. Maintaining such structures enhances stability, accuracy, and efficiency. Linearization, a technique that reformulates rational matrix problems as generalized eigenvalue problems (GEPs) of larger matrices, is widely used but does not automatically retain structure. In this chapter, we focus on structured linearizations, which preserve both the spectral information of the original rational matrix and its intrinsic structural properties. To achieve this, we present the construction of a family of linearizations called generalized Fiedler pencils with repetition (GFPR), which we prove to be valid linearizations for rational matrices. Moreover, we demonstrate that the GFPR family serves as a versatile framework for generating structured linearizations, specifically symmetric, skew-symmetric, T-even, and T-odd linearizations, provided the original rational matrix exhibits the corresponding structure. These structured linearizations facilitate the use of specialized, structure-preserving algorithms, reduce numerical errors, and yield physically meaningful solutions in application |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_21211 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Structured Linearizations of Structured Rational Matrices Bist, Avisek Behera, Namita Rings and Algebras 65F15, 15A57, 15A18, 65F35 Numerical computations involving rational matrices often benefit from preserving underlying matrix structures such as symmetry, Hermitian properties, or sparsity that reflect physical, geometric, or algebraic characteristics of the system. Maintaining such structures enhances stability, accuracy, and efficiency. Linearization, a technique that reformulates rational matrix problems as generalized eigenvalue problems (GEPs) of larger matrices, is widely used but does not automatically retain structure. In this chapter, we focus on structured linearizations, which preserve both the spectral information of the original rational matrix and its intrinsic structural properties. To achieve this, we present the construction of a family of linearizations called generalized Fiedler pencils with repetition (GFPR), which we prove to be valid linearizations for rational matrices. Moreover, we demonstrate that the GFPR family serves as a versatile framework for generating structured linearizations, specifically symmetric, skew-symmetric, T-even, and T-odd linearizations, provided the original rational matrix exhibits the corresponding structure. These structured linearizations facilitate the use of specialized, structure-preserving algorithms, reduce numerical errors, and yield physically meaningful solutions in application |
| title | Structured Linearizations of Structured Rational Matrices |
| topic | Rings and Algebras 65F15, 15A57, 15A18, 65F35 |
| url | https://arxiv.org/abs/2602.21211 |