Journal of Mathematical Sciences

, Volume 243, Issue 1, pp 45–55 | Cite as

On the Divisibility of Matrices with Remainder over the Domain of Principal Ideals

  • V. М. Prokip

We study the problem of divisibility of matrices with remainder over a domain of principal ideals R and establish the conditions under which, for a pair of (n × n)-matrices A and B over the domain R , there exists a unique pair of (n × n)-matrices P and Q over R such that B = AP +Q. The application of the obtained results to finding special solutions of a Sylvester-type matrix equation is presented.


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  1. 1.
    F. R. Gantmacher, The Theory of Matrices, Vol. 1, Chelsea Publ. Co., New York (1959).zbMATHGoogle Scholar
  2. 2.
    N. Dzhalyuk and V. Petrychkovych, “Semiscalar equivalence of polynomial matrices and the solution of Sylvester matrix polynomial equations,” Mat. Visn. NTSh,9, 81–88 (2013).zbMATHGoogle Scholar
  3. 3.
    O. Zariski and P. Samuel, Commutative Algebra, Vol. 1, Van Nostrand, Princeton, NJ (1958).zbMATHGoogle Scholar
  4. 4.
    G. V. Kalaidzhich, “Euclidean algorithm in matrix modules over a given Euclidean ring,” Sib. Mat. Zh.,26, No. 6, 48–53 (1985); English translation:Sib. Math. J.,26, Issue 6, 818–822 (1985).MathSciNetCrossRefGoogle Scholar
  5. 5.
    N. B. Ladzoryshyn, “Integer solutions of matrix linear unilateral and bilateral equations over quadratic rings,” Mat. Met. Fiz.-Mekh. Polya,58, No. 2, 47–54 (2015); English translation:J. Math. Sci.,223, No. 1, 50–59 (2017).MathSciNetCrossRefGoogle Scholar
  6. 6.
    V. M. Prokip, “On the solvability of a system of linear equations over the domain of principal ideals,” Ukr. Mat. Zh.,66, No. 4, 566–570 (2014); English translation:Ukr. Math. J.,66, No. 4, 633–637 (2014).MathSciNetCrossRefGoogle Scholar
  7. 7.
    K. A. Rodosskii, Euclidean Algorithm [in Russian], Nauka, Moscow (1988).zbMATHGoogle Scholar
  8. 8.
    I. N. Sanov, “Euclidean algorithm and one-sided prime factorization for matrix rings,” Sib. Mat. Zh.,8, No. 4, 846–852 (1967).MathSciNetzbMATHGoogle Scholar
  9. 9.
    S. Barnett, Polynomials and Linear Control Systems, Marcel Dekker, New York, (1983).zbMATHGoogle Scholar
  10. 10.
    S. Barnett, “Regular polynomial matrices having relatively prime determinants,” Math. Proc. Cambridge Phil. Soc.,65, No. 3, 585–590 (1969).MathSciNetCrossRefGoogle Scholar
  11. 11.
    H.-H. Brungs, “Left Euclidean rings,” Pacific J. Math.,45, No. 1, 27–33 (1973).MathSciNetCrossRefGoogle Scholar
  12. 12.
    P. M. Cohn, Free Rings and Their Relations, Academic Press, London (1985).zbMATHGoogle Scholar
  13. 13.
    S. Chen and Y. Tian, “On solutions of generalized Sylvester equation in polynomial matrices,” J. Franklin Inst.,351, No. 12, 5376–5385 (2014).MathSciNetCrossRefGoogle Scholar
  14. 14.
    R. B. Feinberg, “Equivalence of partitioned matrices,” J. Res. Nat. Bur. Stand. Sect. B: Math. Sci.,80B, No. 1, 89–98 (1976).MathSciNetCrossRefGoogle Scholar
  15. 15.
    J. Feinstein and Y. Barness, “On the uniqueness minimal solution of the matrix polynomial equation A(λ)X(λ) + Y (λ)B(λ) = C(λ),” J. Franklin Inst.,310, No. 2, 131–134 (1980).MathSciNetCrossRefGoogle Scholar
  16. 16.
    C. R. Fletcher, “Euclidean rings,” J. London Math. Soc.,s2-4, No. 1, 79–82. (1971).MathSciNetCrossRefGoogle Scholar
  17. 17.
    W. H. Gustafson, “Roth’s theorems over commutative rings,” Linear Algebra Appl.,23, 245–251 (1979).MathSciNetCrossRefGoogle Scholar
  18. 18.
    A. V. Jategaonkar, “Rings with transfinite left division algorithm,” Bull. Amer. Math. Soc.,75, No. 3, 559–561 (1969).MathSciNetCrossRefGoogle Scholar
  19. 19.
    M. A. Kaashoek and L. Lerer, “On a class of matrix polynomial equations,” Linear Algebra Appl.,439, No. 3, 613–620 (2013).MathSciNetCrossRefGoogle Scholar
  20. 20.
    T. Kaczorek, Polynomial and Rational Matrices. Applications in Dynamical Systems Theory, Springer, London (2007).CrossRefGoogle Scholar
  21. 21.
    P. Lezowski, “On some Euclidean properties of matrix algebras,” J. Algebra,486, 157–203 (2017).MathSciNetCrossRefGoogle Scholar
  22. 22.
    T. Motzkin, “The Euclidean algorithm,” Bul. Amer. Math. Soc.,55, No. 12, 1142–1146 (1949).MathSciNetCrossRefGoogle Scholar
  23. 23.
    M. Newman, “The Smith normal form of a partitioned matrix,” J. Res. Nat. Bur. Stand. Sect. B: Math. Sci.,78B, No. 1, 3–6 (1974).MathSciNetCrossRefGoogle Scholar
  24. 24.
    M. Newman, Integral Matrices, Ser. Pure and Applied Mathematics, Vol. 45, Academic Press, New York (1972).Google Scholar
  25. 25.
    V. Olshevsky, “Similarity of block diagonal and block triangular matrices,” Integr. Equat. Oper. Theory,15, No. 5, 853–863 (1992).MathSciNetCrossRefGoogle Scholar
  26. 26.
    V. M. Prokip, “About the uniqueness solution of the matrix polynomial equation A(λ)X(λ) + Y (λ)B(λ) = C(λ),” Lobachevskii J. Math.,29, No. 3, 186–191 (2008).MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  • V. М. Prokip
    • 1
  1. 1.Pidstryhach Institute for Applied Problems in Mechanics and Mathematics, Ukrainian National Academy of SciencesLvivUkraine

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