About this Journal Submit a Manuscript Table of Contents
ISRN Algebra
Volume 2012 (2012), Article ID 205478, 14 pages
http://dx.doi.org/10.5402/2012/205478
Research Article

The Matrix Linear Unilateral and Bilateral Equations with Two Variables over Commutative Rings

Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, National Academy of Sciences of Ukraine, 3-b, Naukova Street, 79060 L'viv, Ukraine

Received 16 January 2012; Accepted 20 February 2012

Academic Editors: I. Cangul, H. Chen, and P. Damianou

Copyright © 2012 N. S. Dzhaliuk and V. M. Petrychkovych. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. T. Kaczorek, Polynomial and Rational Matrices, Communications and Control Engineering Series, Springer, London, UK, 2007. View at Zentralblatt MATH
  2. V. Kučera, “Algebraic theory of discrete optimal control for single-variable systems. I. Preliminaries,” Kybernetika, vol. 9, pp. 94–107, 1973. View at Zentralblatt MATH
  3. V. Kučera, “Algebraic theory of discrete optimal control for multivariable systems,” Kybernetika, vol. 10/12, supplement, pp. 3–56, 1974.
  4. W. A. Wolovich and P. J. Antsaklis, “The canonical Diophantine equations with applications,” SIAM Journal on Control and Optimization, vol. 22, no. 5, pp. 777–787, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. W. E. Roth, “The equations AXYB=C and AXXB=C in matrices,” Proceedings of the American Mathematical Society, vol. 3, pp. 392–396, 1952. View at Zentralblatt MATH
  6. M. Newman, “The Smith normal form of a partitioned matrix,” Journal of Research of the National Bureau of Standards, vol. 78, pp. 3–6, 1974. View at Zentralblatt MATH
  7. C. R. Johnson and M. Newman, “A condition for the diagonalizability of a partitioned matrix,” Journal of Research of the National Bureau of Standards, vol. 79, no. 1-2, pp. 45–48, 1975. View at Zentralblatt MATH
  8. R. B. Feinberg, “Equivalence of partitioned matrices,” Journal of Research of the National Bureau of Standards, vol. 80, no. 1, pp. 89–97, 1976. View at Zentralblatt MATH
  9. W. H. Gustafson, “Roth's theorems over commutative rings,” Linear Algebra and Its Applications, vol. 23, pp. 245–251, 1979. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. R. E. Hartwig, “Roth's equivalence problem in unit regular rings,” Proceedings of the American Mathematical Society, vol. 59, no. 1, pp. 39–44, 1976. View at Zentralblatt MATH
  11. W. H. Gustafson and J. M. Zelmanowitz, “On matrix equivalence and matrix equations,” Linear Algebra and Its Applications, vol. 27, pp. 219–224, 1979. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. R. M. Guralnick, “Roth's theorems and decomposition of modules,” Linear Algebra and Its Applications, vol. 39, pp. 155–165, 1981. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. L. Huang and J. Liu, “The extension of Roth's theorem for matrix equations over a ring,” Linear Algebra and Its Applications, vol. 259, pp. 229–235, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. R. E. Hartwig and P. Patricio, “On Roth's pseudo equivalence over rings,” Electronic Journal of Linear Algebra, vol. 16, pp. 111–124, 2007. View at Zentralblatt MATH
  15. P. Lancaster and M. Tismenetsky, The Theory of Matrices, Computer Science and Applied Mathematics, Academic Press, Orlando, Fla, USA, 2nd edition, 1985.
  16. S. Barnett, “Regular polynomial matrices having relatively prime determinants,” Proceedings of the Cambridge Philosophical Society, vol. 65, pp. 585–590, 1969. View at Zentralblatt MATH
  17. V. Petrychkovych, “Cell-triangular and cell-diagonal factorizations of cell-triangular and cell-diagonal polynomial matrices,” Mathematical Notes, vol. 37, no. 6, pp. 431–435, 1985.
  18. J. Feinstein and Y. Bar-Ness, “On the uniqueness of the minimal solution to the matrix polynomial equation A(λ)X(λ)+Y(λ)B(λ)=C(λ),” Journal of the Franklin Institute, vol. 310, no. 2, pp. 131–134, 1980. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. V. M. Prokip, “About the uniqueness solution of the matrix polynomial equation A(λ)X(λ)Y(λ)B(λ)=C(λ),” Lobachevskii Journal of Mathematics, vol. 29, no. 3, pp. 186–191, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. V. Petrychkovych, “Generalized equivalence of pairs of matrices,” Linear and Multilinear Algebra, vol. 48, no. 2, pp. 179–188, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  21. V. Petrychkovych, “Standard form of pairs of matrices with respect to generalized eguivalence,” Visnyk of Lviv University, vol. 61, pp. 153–160, 2003.
  22. P. M. Cohn, Free Rings and Their Relations, Academic Press, London, UK, 1971.
  23. S. Friedland, “Matrices over integral domains,” in CRC Handbook of Linear Algebra, pp. 23-1–23-11, Chapman & Hall, New York, NY, USA, 2007.
  24. M. Newman, Integral Matrices, Academic Press, New York, NY, USA, 1972.
  25. B. L. Van der Waerden, Algebra, Springer, New York, NY, USA, 1991.
  26. A. I. Borevich and I. R. Shafarevich, Number Theory, vol. 20 of Translated from the Russian by Newcomb Greenleaf. Pure and Applied Mathematics, Academic Press, New York, NY, USA, 1966.
  27. K. A. Rodossky, Euclid's Algorithm, Nauka, Moscow, Russia, 1988.
  28. I. Kaplansky, “Elementary divisors and modules,” Transactions of the American Mathematical Society, vol. 66, pp. 464–491, 1949. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  29. O. Helmer, “The elementary divisor theorem for certain rings without chain condition,” Bulletin of the American Mathematical Society, vol. 49, pp. 225–236, 1943. View at Publisher · View at Google Scholar · View at Zentralblatt MATH