Bessmertnyĭ realizations of symmetric multivariate rational matrix functions over any field.
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| Title: | Bessmertnyĭ realizations of symmetric multivariate rational matrix functions over any field. |
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| Authors: | Elsinger, Jason1 (AUTHOR), Orzel, Ian2 (AUTHOR), Welters, Aaron1,3 (AUTHOR) awelters@fit.edu |
| Source: | Linear Algebra & its Applications. May2026, Vol. 737, p80-118. 39p. |
| Subjects: | Schur complement, State-space methods, Systems theory |
| Abstract: | In this paper, we prove the following. First, every square matrix whose entries are multivariable rational functions over a field F has a Bessmertnyĭ realization, i.e., is the Schur complement of an affine linear square matrix pencil with coefficients in F. Second, if the matrix is also symmetric and the characteristic of the field F is not two then it has a symmetric Bessmertnyĭ realization (i.e., the pencil can be chosen to consist of symmetric matrices) and counterexamples are given to prove this statement is false in general for fields of characteristic two. Third, for fields of characteristic two (e.g., binary or Boolean field), we completely characterize those functions that have a symmetric Bessmertnyĭ realization. Finally, analogous results hold when restricted to the class of homogeneous degree-one rational functions. To solve these realization problems, i.e., finding such structured Bessmertnyĭ realizations for a given multivariate rational function, we use state-space methods from systems theory to produce realizations for algebraic operations on Schur complements such as sums, products, inverses, and symmetrization, which become the elementary building blocks of our constructions. Further complications arise over fields of characteristic two, so a large part of the paper is devoted to developing additional methods to decide if the symmetric realization problem can be solved and, if so, to construction the symmetric realization for a given symmetric rational matrix function. Our motivations are discussed in the context of multidimensional linear systems theory on generalizing state-space representations for rational functions including the Givone-Roesser and Fornasini-Marchesini realizations. [ABSTRACT FROM AUTHOR] |
| Copyright of Linear Algebra & its Applications is the property of Elsevier B.V. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
| Database: | Engineering Source |
| FullText | Text: Availability: 0 |
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| Header | DbId: egs DbLabel: Engineering Source An: 192196633 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
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| Items | – Name: Title Label: Title Group: Ti Data: Bessmertnyĭ realizations of symmetric multivariate rational matrix functions over any field. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Elsinger%2C+Jason%22">Elsinger, Jason</searchLink><relatesTo>1</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Orzel%2C+Ian%22">Orzel, Ian</searchLink><relatesTo>2</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Welters%2C+Aaron%22">Welters, Aaron</searchLink><relatesTo>1,3</relatesTo> (AUTHOR)<i> awelters@fit.edu</i> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Linear+Algebra+%26+its+Applications%22">Linear Algebra & its Applications</searchLink>. May2026, Vol. 737, p80-118. 39p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Schur+complement%22">Schur complement</searchLink><br /><searchLink fieldCode="DE" term="%22State-space+methods%22">State-space methods</searchLink><br /><searchLink fieldCode="DE" term="%22Systems+theory%22">Systems theory</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: In this paper, we prove the following. First, every square matrix whose entries are multivariable rational functions over a field F has a Bessmertnyĭ realization, i.e., is the Schur complement of an affine linear square matrix pencil with coefficients in F. Second, if the matrix is also symmetric and the characteristic of the field F is not two then it has a symmetric Bessmertnyĭ realization (i.e., the pencil can be chosen to consist of symmetric matrices) and counterexamples are given to prove this statement is false in general for fields of characteristic two. Third, for fields of characteristic two (e.g., binary or Boolean field), we completely characterize those functions that have a symmetric Bessmertnyĭ realization. Finally, analogous results hold when restricted to the class of homogeneous degree-one rational functions. To solve these realization problems, i.e., finding such structured Bessmertnyĭ realizations for a given multivariate rational function, we use state-space methods from systems theory to produce realizations for algebraic operations on Schur complements such as sums, products, inverses, and symmetrization, which become the elementary building blocks of our constructions. Further complications arise over fields of characteristic two, so a large part of the paper is devoted to developing additional methods to decide if the symmetric realization problem can be solved and, if so, to construction the symmetric realization for a given symmetric rational matrix function. Our motivations are discussed in the context of multidimensional linear systems theory on generalizing state-space representations for rational functions including the Givone-Roesser and Fornasini-Marchesini realizations. [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of Linear Algebra & its Applications is the property of Elsevier B.V. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1016/j.laa.2026.02.014 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 39 StartPage: 80 Subjects: – SubjectFull: Schur complement Type: general – SubjectFull: State-space methods Type: general – SubjectFull: Systems theory Type: general Titles: – TitleFull: Bessmertnyĭ realizations of symmetric multivariate rational matrix functions over any field. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Elsinger, Jason – PersonEntity: Name: NameFull: Orzel, Ian – PersonEntity: Name: NameFull: Welters, Aaron IsPartOfRelationships: – BibEntity: Dates: – D: 15 M: 05 Text: May2026 Type: published Y: 2026 Identifiers: – Type: issn-print Value: 00243795 Numbering: – Type: volume Value: 737 Titles: – TitleFull: Linear Algebra & its Applications Type: main |
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