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Ukrainian Mathematical Journal

, Volume 66, Issue 3, pp 479–485 | Cite as

Greatest common divisor of matrices one of which is a disappear matrix

  • A. M. Romaniv
  • V. P. Shchedryk
Article
  • 48 Downloads

We study the structure of the greatest common divisor of matrices one of which is a disappear matrix. In this connection, we indicate the Smith normal form and the transforming matrices of the left greatest common divisor.

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References

  1. 1.
    I. Kaplansky, “Elementary divisor and modules,” Trans. Amer. Math. Soc., 66, 464–491 (1949).CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    C. C. MacDuffee, “Matrices with elements in a principal ring,” Bull. Amer. Math. Soc., 39, 570–573 (1933).Google Scholar
  3. 3.
    E. Cahen, Théorie des Nombres, Vol. 1, Hermann, Paris (1914).Google Scholar
  4. 4.
    A. Châtelet, Groupes Abéliens Finis, Gauthier, Paris (1924).Google Scholar
  5. 5.
    S. Barnett, “Regular greatest common divisor of two polynomial matrices,” Proc. Cambridge Phil. Soc., 72, 161–165 (1972).CrossRefzbMATHGoogle Scholar
  6. 6.
    S. Barnett, “Greatest common divisors from generalized Sylvester resultant matrices,” Lin. Multilin. Algebra, 8, 271–279 (1980).CrossRefzbMATHGoogle Scholar
  7. 7.
    R. R. Bitmead, S. Y. Kung, O. Anderson, and T. Kailath, “Greatest common divisors via generalized Sylvester and Bezout matrices,” IEEE Trans. Automat. Contr., 23, No. 6, 1043–1047 (1978).CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    I. Gohberg, M. A. Kaashoek, L. Lerer, and L. Rodman, “Common multiples and common divisors of matrix polynomials, I,” Indiana Univ. Math. J., 30, No. 3, 321–356 (1981).CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    R. C. Thompson, “Left multiples and right divisors of integral matrices,” Lin. Multilin. Algebra, 19, 287–295 (1986).CrossRefzbMATHGoogle Scholar
  10. 10.
    N. S. Dzhalyuk and V. M. Petrychkovych, “On common unital divisors of polynomial matrices with given canonical diagonal form,” Mat. Met. Fiz.-Mekh. Polya, 45, No. 3, 7–13 (2002).zbMATHMathSciNetGoogle Scholar
  11. 11.
    V. M. Prokip, “On common divisors of matrices over factorial domains,” Mat. Met. Fiz.-Mekh. Polya, 48, No. 4, 43–50 (2005).zbMATHMathSciNetGoogle Scholar
  12. 12.
    V. C. Nanda, “On the gcd and lcm of matrices over Dedekind domains,” in: Number Theory and Discrete Mathematics, Trends in Mathematics (2002), pp. 201–211.Google Scholar
  13. 13.
    V. R. Zelisko, “On the structure of one class of invertible matrices,” Mat. Met. Fiz.-Mekh. Polya, Issue 12, 14–21 (1980).Google Scholar
  14. 14.
    V. Shchedryk, “Factorization of matrices over elementary divisor domain,” Alg. Discrete Math., No. 2, 79–99 (2009).MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • A. M. Romaniv
    • 1
  • V. P. Shchedryk
    • 1
  1. 1.Institute for Applied Problems of Mechanics and MathematicsUkrainian National Academy of SciencesLvivUkraine

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