References

  1. Dzhaliuk N.S., Petrychkovych V.M. Solutions of the matrix linear bilateral polynomial equation and their structure. Algebra Discrete Math. 2019, 27 (2), 243-251.
  2. Dzhaliuk N.S., Petrychkovych V.M. The matrix linear unilateral and bilateral equations with two variables over commutative rings. ISRN Algebra 2012, Article ID 205478, 14 pages. doi: 10.5402/2012/205478
  3. Dzhaliuk N., Petrychkovych V. The semiscalar equivalence of polynomial matrices and the solution of the Sylvester matrix polynomial equations. Math. Bull. Shevchenko Sci. Soc. 2012, 9, 81-88. (in Ukrainian)
  4. Feinstein J., Bar-Ness Y. On the uniqueness of the minimal solution the matrix polynomial equation $A(\lambda)X(\lambda)+Y(\lambda)B(\lambda)=C(\lambda)$. J. Franklin Inst. 1980, 310 (2), 131-134. doi: 10.1016/0016-0032(78)90012-1
  5. Kaczorek T. Polynomial and Rational Matrices: applications in dynamical system theory. In: Commun. and Control Eng. Springer-Verlag, London, 2007. doi: 10.1007/978-1-84628-605-6
  6. Kazimirs'kii P.S. Factorization of matrix polynomials. Pidstryhach Inst. for Appl. Probl. of Mech. and Math. of the NAS of Ukraine, L'viv, 2015. (in Ukrainian)
  7. Kučera V. Algebraic theory of discrete optimal control for single-variable systems. I. Preliminaries. Kybernetika (Prague) 1973, 9 (2), 94-107.
  8. Ladzoryshyn N.B. On equivalence of pairs of matrices, in which determinants are orders of primes, over quadratic Euclidean rings. Carpathian Math. Publ. 2013, 5 (1), 63-69. doi: 10.15330/cmp.5.1.63-69 (in Ukrainian)
  9. Ladzoryshyn N.B. Integer Solutions of Matrix Linear Unilateral and Bilateral Equations over Quadratic Rings. J. Math. Sci. 2017, 223 (1), 50-59. doi: 10.1007/s10958-017-3337-0 (translation of Mat. Metody Fiz.-Mekh. Polya 2015, 58 (2), 47-54. (in Ukrainian))
  10. Ladzoryshyn N., Petrychkovych V. Equivalence of pairs of matrices with relatively prime determinants over quadratic rings of principal ideals. Bul. Acad. Ştiinţe Repub. Mold. Mat. 2014, 3, 38-48.
  11. Ladzoryshyn N., Petrychkovych V. Matrix linear unilateral and bilateral equations over quadratic rings. Visnyk of the Lviv Univ. Ser. Mech. Math. 2018, 85, 32-40. doi: 10.30970/vmm.2018.85.032-040 (in Ukrainian)
  12. Ladzoryshyn N.B., Petrychkovych V.M. The standard form of matrices over quadratic rings with respect to $(z,k)$-equivalence and structure of solutions of matrix linear bilateral equations. Mat. Metody Fiz.-Mekh. Polya 2018, 61 (2), 49-56. (in Ukrainian)
  13. Lancaster P., Tismenetsky M. The Theory of Matrices. Computer Science and Applied Mathematics, Academic Press, Orlando, Fla, USA, 2nd edition, 1985.
  14. Petrichkovich V. M. Semiscalar equivalence and the Smith normal form of polynomial matrices. J. Math. Sci. 1993, 66 (1), 2030-2033. doi: 10.1007/BF01097386 (translation of Mat. Metodi Fiz.-Mekh. Polya 1987, 26, 13--16. (in Russian))
  15. Petrychkovych V.M. Generalized equivalence of matrices and its collections and factorization of matrices over rings. Pidstryhach Inst. Appl. Probl. Mech. and Math. of the NAS of Ukraine, L'viv, 2015. (in Ukrainian)
  16. Petrychkovych V. Generalized equivalence of pairs of matrices. Linear Multilinear Algebra 2000, 48 (2), 179-188. doi: 10.1080/03081080008818667
  17. Petrychkovych V. Standard form of pairs of matrices with respect to generalized equivalence. Visnyk of the Lviv Univ. Ser. Mech. Math. 2003, 61, 148-155. (in Ukrainian)
  18. Tzekis P.A. A new algorithm for the solution of a polynomial matrix Diophantine equation. Appl. Math. Comput. 2007, 193(2), 395-407. doi: 10.1016/j.amc.2007.03.076
  19. Wolovich W.A., Antsaklis P.J. The canonical Diophantine equations with applications. SIAM J. Control Optim. 1984, 22 (5), 777-787. doi: 10.1137/0322049
  20. Zelisko V.R., Ladzoryshyn N.B., Petrychkovych V.M. On equivalence of matrices over quadratic Euclidean rings. Appl. Probl. Mech. Math. 2006, 4, 16-21. (in Ukrainian)
  21. Zhou B., Yan Z.B., Duan G.R. Unified Parametrization for the Solutions to the Polynomial Diophantine Matrix Equation and the Generalized Sylvester Matrix Equation. Int. J. of Control, Automation, and Syst. 2010, 8 (1), 29-35. doi: 10.1007/s12555-010-0104-0