On the criteria for extracting a factor from a matrix polynomia

V. S. Ilkiv; Z. M. Nytrebych

doi:10.30970/ms.66.1.3-10

V. S. Ilkiv Lviv Polytechnic National University, Department of Mathematics Lviv, Ukraine
Z. M. Nytrebych Lviv Polytechnic National University, Department of Mathematics Lviv, Ukraine https://orcid.org/0000-0002-9599-8517

DOI: https://doi.org/10.30970/ms.66.1.3-10

Keywords: matrix polynomial, matrix factorization, monic factor, factorization criteria

Abstract

The problem of extracting a monic factor with a prescribed determinant from a matrix polynomial, i.e., the factorization problem, is studied. Based on Kazimirskii's well-known criterion, new factorization criteria using only the coefficients of the matrix polynomial and those of the prescribed determinant are established. An algorithm for determining such a factor is also proposed, and examples illustrating one of the criteria are presented. New forms of a criterion for extracting a monic factor with a given determinant from a matrix polynomial have been obtained. The algorithm for calculating the coefficients of the desired factor, which consists in solving a (reduced) system of matrix equations, does not involve finding the roots of the characteristic polynomial or the adjugate matrix. It uses only the coefficients of the given matrix polynomial and the coefficients of the determinant of the factor.

Author Biographies

V. S. Ilkiv, Lviv Polytechnic National University, Department of Mathematics Lviv, Ukraine

Lviv Polytechnic National University, Department of Mathematics
Lviv, Ukraine

Z. M. Nytrebych, Lviv Polytechnic National University, Department of Mathematics Lviv, Ukraine

Lviv Polytechnic National University, Department of Mathematics
Lviv, Ukraine

References

Ya.B. Lopatinskij, Decomposition of a Polynomial Matrix into a Product, Nauchn. Zap. L’vov. Polytechn. Inst., Ser. Fiz. Mat. 38 (2) (1953), 3–7.

Ya.B. Lopatinskij, On Some Properties of Polynomial Matrices, Boundary Value Probl. of Math. Phys., Inst. of Math. AN USSR, Kiev, 1979, 108–116.

P.S. Kazimirs’kii, To the Decomposition of a Polynomial Matrix into Linear Factors, Dopovidi AN URSR 4 (1964), 446–448.

V. Shchedryk, Arithmetic of Matrices over Rings, Akademperiodyka, Kyiv, 2021. https://doi.org/10.15407/akademperiodika.430.278

P.S. Kazimirs’kii, Factorization of Matrix Polynomials, Pidstryhach Inst. for Appl. Probl. of Mech. and Math. of the NAS of Ukraine, Lviv, 2015.

I. Gohberg, P. Lancaster, L. Rodman, Matrix Polynomials, Academic Press, New York, 1982.

V.M. Petrychkovych, Generalized Equivalence of Matrices and Its Collections and Factorization of Matrices over Rings, Pidstryhach Inst. for Appl. Probl. of Mech. and Math. of the NAS of Ukraine, Lviv, 2015.

V.P. Shchedryk, Factorization of Matrices over Elementary Divisor Domain, Algebra Discrete Math. (2) (2009), 79–99.

V. Ilkiv, On the Criterion for Extracting a Factor of a Matrix Polynomial, 15th Ukraine algebra conference: book of abstracts, Lviv, July 8–12, 2025, p. 58.

V.S. Il’kiv, New Forms of the Lopatynsky Condition, Ukr. Math. J. 77 (2) (2025), 107–122. https://doi.org/10.3842/umzh.v77i2.8534

V.S. Ilkiv, P.I. Kalenyuk, Verification Algorithm for Lopatynsky Condition, Mathematical Modeling and Computing 11 (4) (2024), 1189–1197. https://doi.org/10.23939/mmc2024.04.1189

Ya.B. Lopatinskij, On a Method of Reducing Boundary Problems for a System of Differential Equations of Elliptic Type to Regular Integral Equations, Ukr. Math. J. 5 (2) (1953), 123–151; Engl. transl., Amer. Math. Transl. (2) 89 (1970), 149–183.

V.S. Il’kiv, B.Yo. Ptashnyk, Problems for Partial Differential Equations with Nonlocal Conditions. Metric Approach to the Problem of Small Denominators, Ukr. Math. J. 58 (12) (2006), 1847–1875. https://doi.org/10.1007/s11253-006-0172-8

B.Yo. Ptashnyk, V.S. Il’kiv, I.Ya. Kmit’, V.M. Polishchuk, Nonlocal Boundary-Value Problems for Partial Differential Equations, Naukova Dumka, Kyiv, 2002.