Triangular form of Laurent polynomial matrices and their factorization

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dc.citation.epage129
dc.citation.issue1
dc.citation.spage119
dc.contributor.affiliationНаціональний університет “Львівська політехніка”
dc.contributor.affiliationЛьвівський національний університет ім. Івана Франка
dc.contributor.affiliationLviv Polytechnic National University
dc.contributor.affiliationIvan Franko National University of Lviv
dc.contributor.authorКучма, М. І.
dc.contributor.authorГаталевич, А. І.
dc.contributor.authorKuchma, M. I.
dc.contributor.authorGatalevych, A. I.
dc.coverage.placenameЛьвів
dc.coverage.placenameLviv
dc.date.accessioned2023-12-13T09:10:59Z
dc.date.available2023-12-13T09:10:59Z
dc.date.created2021-03-01
dc.date.issued2021-03-01
dc.description.abstractДосліджено питання напіскалярної еквівалентності поліноміальних матриць Лорана і встановлена відносно цієї еквівалентності трикутна форма таких матриць та їх скінченних наборів. Доведено теорему про регуляризацію для поліноміальних матриць Лорана. Ця теорема використовується у задачі факторизації таких матриць. Отримано критерій факторизації поліноміальних матриць Лорана із регулярним множником із наперед заданою нормальною формою Сміта.
dc.description.abstractThe issue of the semiscalar equivalence of Laurent polynomial matrices is investigated and the triangular form of such matrices and their finite sets is established with respect to this equivalence. The theorem on regularization of a Laurent polynomial matrix is proved. This theorem is used in the problem of factorization of such matrices. The factorization criterion of a Laurent polynomial matrix with a regular multiplier with a predetermined Smith normal form is obtained.
dc.format.extent119-129
dc.format.pages11
dc.identifier.citationKuchma M. I. Triangular form of Laurent polynomial matrices and their factorization / M. I. Kuchma, A. I. Gatalevych // Mathematical Modeling and Computing. — Lviv : Lviv Politechnic Publishing House, 2022. — Vol 9. — No 1. — P. 119–129.
dc.identifier.citationenKuchma M. I. Triangular form of Laurent polynomial matrices and their factorization / M. I. Kuchma, A. I. Gatalevych // Mathematical Modeling and Computing. — Lviv : Lviv Politechnic Publishing House, 2022. — Vol 9. — No 1. — P. 119–129.
dc.identifier.doi10.23939/mmc2022.01.119
dc.identifier.urihttps://ena.lpnu.ua/handle/ntb/60542
dc.language.isoen
dc.publisherВидавництво Львівської політехніки
dc.publisherLviv Politechnic Publishing House
dc.relation.ispartofMathematical Modeling and Computing, 1 (9), 2022
dc.relation.references[1] Fornasini E., Valcher M.-E. nD Polynomial Matrices with Applications to Multidimensional Signal Analysis. Multidimensional Systems and Signal Processing. 8 (4), 387–408 (1997).
dc.relation.references[2] Kaczorek T. Polynomial and Rational Matrices: Applications in Dynamical System Theory. Commun. and Control Eng. Ser.; London (UK), Springer, (2007).
dc.relation.references[3] Foster J. A., McWhirter J. G., Davies M. R., Chambers J. A. An algorithm for calculating the QR and singular value decompositions of polynomial matrices. AIEEE Trans. Signal Process. 58 (3), 1263–1274 (2010).
dc.relation.references[4] Park I. Symbolic computation and signal processing. Journal of Symbolic Computation. 37, 209–226 (2004).
dc.relation.references[5] McWhirter J. G., Baxter P. D., Cooper T., Redif S., Foster J. An EVD algo-rithm for Para-Hermitian polynomial matrices. IEEE Trans. Signal Process. 55 (6), 2158–2169 (2007).
dc.relation.references[6] Kazimirskii P. S., Petrychkovych V. M. On the equivalence of polynomial matrices. Theoretical and Applied Problems of Algebra and Differential Equations. Lviv, 61–66 (1977).
dc.relation.references[7] Petrychkovych V. M. On semiscalar equivalence and the Smith normal form of polynomial matrices. Journal of Mathematical Sciences. 66 (1), 2030–2033 (1993).
dc.relation.references[8] Petrychkovych V. M. Generalized equivalence of pair of matrices. Linear Multi-linear Algebra. 48, 179–188 (2000).
dc.relation.references[9] Petrychkovych V. M. Standart form of pair of matrices with respect to generalized equivalence. Visnyk Lviv. Univ. 61, 153–160 (2003).
dc.relation.references[10] Kuchma M. I. Symmetric equivalence of matrix polynomials and isolation of a common unital divisor in matrix polynomials. Ukrainian Mathematical Journal. 53 (2), 238–248 (2001).
dc.relation.references[11] Dias da Silva J. A., Laffey T. A. On simultaneous similarity of matrices and related questions. Linear Algebra Appl. 291, 167–184 (1999).
dc.relation.references[12] Kazimirskii P. S. Factorization of matrix polynomials. Lviv, Pidstryhach Institute for Applied Problems of Mechanics and Mathematics of the NAS of Ukraine, 2-nd edition, (2015), 282 p.
dc.relation.references[13] Zelisko V. R., Kuchma M. I. Factorization of symmetric matrices over polynomial rings with involution. Journal of Mathematical Sciences. 96, 3017–3021 (1999).
dc.relation.references[14] Zelisko V. R., Shchedryk V. P. Matrix of values on a system of roots of diagonal elements of matrix and its applications. Mat. Met. i Fiz.-Mekh. Polya. 48 (4), 20–29 (2005).
dc.relation.references[15] Shchedryk V. P. Arithmetic of matrices over rings. Kyiv, Akademperiodyka (2021), 278 p.
dc.relation.references[16] Kazimirskiy P. S., Shchedryk V. P. On solutions of matrix polynomials sides equations. Doklady AN SSSR. 304 (2), 271–274 (1989).
dc.relation.referencesen[1] Fornasini E., Valcher M.-E. nD Polynomial Matrices with Applications to Multidimensional Signal Analysis. Multidimensional Systems and Signal Processing. 8 (4), 387–408 (1997).
dc.relation.referencesen[2] Kaczorek T. Polynomial and Rational Matrices: Applications in Dynamical System Theory. Commun. and Control Eng. Ser.; London (UK), Springer, (2007).
dc.relation.referencesen[3] Foster J. A., McWhirter J. G., Davies M. R., Chambers J. A. An algorithm for calculating the QR and singular value decompositions of polynomial matrices. AIEEE Trans. Signal Process. 58 (3), 1263–1274 (2010).
dc.relation.referencesen[4] Park I. Symbolic computation and signal processing. Journal of Symbolic Computation. 37, 209–226 (2004).
dc.relation.referencesen[5] McWhirter J. G., Baxter P. D., Cooper T., Redif S., Foster J. An EVD algo-rithm for Para-Hermitian polynomial matrices. IEEE Trans. Signal Process. 55 (6), 2158–2169 (2007).
dc.relation.referencesen[6] Kazimirskii P. S., Petrychkovych V. M. On the equivalence of polynomial matrices. Theoretical and Applied Problems of Algebra and Differential Equations. Lviv, 61–66 (1977).
dc.relation.referencesen[7] Petrychkovych V. M. On semiscalar equivalence and the Smith normal form of polynomial matrices. Journal of Mathematical Sciences. 66 (1), 2030–2033 (1993).
dc.relation.referencesen[8] Petrychkovych V. M. Generalized equivalence of pair of matrices. Linear Multi-linear Algebra. 48, 179–188 (2000).
dc.relation.referencesen[9] Petrychkovych V. M. Standart form of pair of matrices with respect to generalized equivalence. Visnyk Lviv. Univ. 61, 153–160 (2003).
dc.relation.referencesen[10] Kuchma M. I. Symmetric equivalence of matrix polynomials and isolation of a common unital divisor in matrix polynomials. Ukrainian Mathematical Journal. 53 (2), 238–248 (2001).
dc.relation.referencesen[11] Dias da Silva J. A., Laffey T. A. On simultaneous similarity of matrices and related questions. Linear Algebra Appl. 291, 167–184 (1999).
dc.relation.referencesen[12] Kazimirskii P. S. Factorization of matrix polynomials. Lviv, Pidstryhach Institute for Applied Problems of Mechanics and Mathematics of the NAS of Ukraine, 2-nd edition, (2015), 282 p.
dc.relation.referencesen[13] Zelisko V. R., Kuchma M. I. Factorization of symmetric matrices over polynomial rings with involution. Journal of Mathematical Sciences. 96, 3017–3021 (1999).
dc.relation.referencesen[14] Zelisko V. R., Shchedryk V. P. Matrix of values on a system of roots of diagonal elements of matrix and its applications. Mat. Met. i Fiz.-Mekh. Polya. 48 (4), 20–29 (2005).
dc.relation.referencesen[15] Shchedryk V. P. Arithmetic of matrices over rings. Kyiv, Akademperiodyka (2021), 278 p.
dc.relation.referencesen[16] Kazimirskiy P. S., Shchedryk V. P. On solutions of matrix polynomials sides equations. Doklady AN SSSR. 304 (2), 271–274 (1989).
dc.rights.holder© Національний університет “Львівська політехніка”, 2022
dc.subjectполіноміальна матриця Лорана
dc.subjectнапівскалярна еквівалентність
dc.subjectтрикутна форма
dc.subjectнормальна форма Сміта
dc.subjectфакторизація матриць
dc.subjectLaurent polynomial matrix
dc.subjectsemiscalar equivalence
dc.subjecttriangular form
dc.subjectSmith normal form
dc.subjectmatrix factorization
dc.titleTriangular form of Laurent polynomial matrices and their factorization
dc.title.alternativeТрикутна форма поліноміальних матриць Лорана та їх факторизація
dc.typeArticle

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Mathematical Modeling And Computing. – 2022. – Vol. 9, No. 1