@article{Ladzoryshyn_Petrychkovych_Zelisko_2020, title={Matrix Diophantine equations over quadratic rings and their solutions}, volume={12}, url={https://journals.pnu.edu.ua/index.php/cmp/article/view/4069}, DOI={10.15330/cmp.12.2.368-375}, abstractNote={<p>The method for solving the matrix Diophantine equations over quadratic rings is developed. On the basic of the standard form of matrices over quadratic rings with respect to $(z,k)$-equivalence previously established by the authors, the matrix Diophantine equation is reduced to equivalent matrix equation of same type with triangle coefficients. Solving this matrix equation is reduced to solving a system of linear equations that contains linear Diophantine equations with two variables, their solution methods are well-known. The structure of solutions of matrix equations is also investigated. In particular, solutions with bounded Euclidean norms are established. It is shown that there exists a finite number of such solutions of matrix equations over Euclidean imaginary quadratic rings. An effective method of constructing of such solutions is suggested.</p>}, number={2}, journal={Carpathian Mathematical Publications}, author={Ladzoryshyn, N.B. and Petrychkovych, V.M. and Zelisko, H.V.}, year={2020}, month={Dec.}, pages={368–375} }