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Volume 2012 (2012), Article ID 205478, 14 pages
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.
- T. Kaczorek, Polynomial and Rational Matrices, Communications and Control Engineering Series, Springer, London, UK, 2007.
- V. Kučera, “Algebraic theory of discrete optimal control for single-variable systems. I. Preliminaries,” Kybernetika, vol. 9, pp. 94–107, 1973.
- V. Kučera, “Algebraic theory of discrete optimal control for multivariable systems,” Kybernetika, vol. 10/12, supplement, pp. 3–56, 1974.
- 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.
- W. E. Roth, “The equations and in matrices,” Proceedings of the American Mathematical Society, vol. 3, pp. 392–396, 1952.
- 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.
- 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.
- R. B. Feinberg, “Equivalence of partitioned matrices,” Journal of Research of the National Bureau of Standards, vol. 80, no. 1, pp. 89–97, 1976.
- W. H. Gustafson, “Roth's theorems over commutative rings,” Linear Algebra and Its Applications, vol. 23, pp. 245–251, 1979.
- 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.
- W. H. Gustafson and J. M. Zelmanowitz, “On matrix equivalence and matrix equations,” Linear Algebra and Its Applications, vol. 27, pp. 219–224, 1979.
- R. M. Guralnick, “Roth's theorems and decomposition of modules,” Linear Algebra and Its Applications, vol. 39, pp. 155–165, 1981.
- 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.
- R. E. Hartwig and P. Patricio, “On Roth's pseudo equivalence over rings,” Electronic Journal of Linear Algebra, vol. 16, pp. 111–124, 2007.
- P. Lancaster and M. Tismenetsky, The Theory of Matrices, Computer Science and Applied Mathematics, Academic Press, Orlando, Fla, USA, 2nd edition, 1985.
- S. Barnett, “Regular polynomial matrices having relatively prime determinants,” Proceedings of the Cambridge Philosophical Society, vol. 65, pp. 585–590, 1969.
- 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.
- J. Feinstein and Y. Bar-Ness, “On the uniqueness of the minimal solution to the matrix polynomial equation ,” Journal of the Franklin Institute, vol. 310, no. 2, pp. 131–134, 1980.
- V. M. Prokip, “About the uniqueness solution of the matrix polynomial equation ,” Lobachevskii Journal of Mathematics, vol. 29, no. 3, pp. 186–191, 2008.
- V. Petrychkovych, “Generalized equivalence of pairs of matrices,” Linear and Multilinear Algebra, vol. 48, no. 2, pp. 179–188, 2000.
- V. Petrychkovych, “Standard form of pairs of matrices with respect to generalized eguivalence,” Visnyk of Lviv University, vol. 61, pp. 153–160, 2003.
- P. M. Cohn, Free Rings and Their Relations, Academic Press, London, UK, 1971.
- S. Friedland, “Matrices over integral domains,” in CRC Handbook of Linear Algebra, pp. 23-1–23-11, Chapman & Hall, New York, NY, USA, 2007.
- M. Newman, Integral Matrices, Academic Press, New York, NY, USA, 1972.
- B. L. Van der Waerden, Algebra, Springer, New York, NY, USA, 1991.
- 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.
- K. A. Rodossky, Euclid's Algorithm, Nauka, Moscow, Russia, 1988.
- I. Kaplansky, “Elementary divisors and modules,” Transactions of the American Mathematical Society, vol. 66, pp. 464–491, 1949.
- O. Helmer, “The elementary divisor theorem for certain rings without chain condition,” Bulletin of the American Mathematical Society, vol. 49, pp. 225–236, 1943.