Dzhaliuk, N. S.; Petrychkovych, V. M. Equivalence of matrices in the ring \(M(n, R)\) and its subrings. (English. Ukrainian original) Zbl 1497.15014 Ukr. Math. J. 73, No. 12, 1865-1872 (2022); translation from Ukr. Mat. Zh. 73, No. 12, 1612-1618 (2021). Summary: We consider the problem of equivalence of matrices in the ring \(M(n, R)\) and its subrings of block triangular matrices \(M_{ BT } (n_1, \dots , n_k, R)\) and block diagonal matrices \(M_{ BD } (n_1, \dots, n_k, R)\), where \(R\) is a commutative domain of principal ideals, and investigate the relationships between these equivalences. Under the condition that the block triangular matrices are block diagonalizable, i.e., equivalent to their main block diagonals, we show that these matrices are equivalent in the subring \(M_{ BT } (n_1, \dots, n_k, R)\) of block triangular matrices if and only if their main diagonals are equivalent in the subring \(M_{ BD } (n_1, \dots , n_k, R)\) of block diagonal matrices, i.e., the corresponding diagonal blocks of these matrices are equivalent. We also prove that if block triangular matrices \(A\) and \(B\) with Smith normal forms \(S(A) = S(B)\) are equivalent to the Smith normal forms in the subring \(M_{ BT } (n_1, \dots, n_k, R)\), then these matrices are equivalent in the subring \(M_{ BT } (n_1, \dots, n_k, R).\) MSC: 15A21 Canonical forms, reductions, classification 15B33 Matrices over special rings (quaternions, finite fields, etc.) Keywords:equivalence of matrices; Smith normal forms PDFBibTeX XMLCite \textit{N. S. Dzhaliuk} and \textit{V. M. Petrychkovych}, Ukr. Math. J. 73, No. 12, 1865--1872 (2022; Zbl 1497.15014); translation from Ukr. Mat. Zh. 73, No. 12, 1612--1618 (2021) Full Text: DOI References: [1] Dmytryshyn, A.; Kågström, B., Coupled Sylvester-type matrix equations and block diagonalization, SIAM J. Matrix Anal. Appl., 36, 2, 580-593 (2015) · Zbl 1328.15028 [2] Roth, WE, The equations AX − YB = C and AX − XB = C in matrices, Proc. Amer. Math. Soc., 3, 392-396 (1952) · Zbl 0047.01901 [3] Feinberg, RB, Equivalence of partitioned matrices, J. Res. Natl. Bur. Stand., 80B, 1, 89-97 (1976) · Zbl 0337.15015 [4] Gustafson, WH, Roth’s theorem over commutative rings, Linear Algebra Appl., 23, 245-251 (1979) · Zbl 0398.15013 [5] Dzhaliuk, NS; Petrychkovych, VM, Solutions of the matrix linear bilateral polynomial equation and their structure, Algebra Discrete Math., 27, 2, 243-251 (2019) · Zbl 1452.15006 [6] V. M. Bondarenko, Representation of Gelfand Graphs [in Ukrainian], Proc. of the Institute of Mathematics, National Academy of Sciences of Ukraine, Kyiv (2005). [7] V. V. Sergeichuk, Canonical Matrices and Related Questions, Proc. of the Institute of Mathematics, National Academy of Sciences of Ukraine, 57, Kyiv (2006). · Zbl 1199.15001 [8] I. Gohberg, P. Lancaster, and L. Rodman, Matrix Polynomials, Academic Press, New York (1982). · Zbl 0482.15001 [9] Petrychkovich, VM, Cell-triangular and cell-diagonal factorizations of cell-triangular and cell-diagonal polynomial matrices, Math. Notes, 37, 6, 431-435 (1985) · Zbl 0591.15009 [10] V. M. Petrychkovych, Generalized Equivalence of Matrices and Their Collections and Factorization of Matrices over Rings [in Ukrainian], Proc. of the Institute for Applied Problems of Mechanics and Mathematics, National Academy of Sciences of Ukraine, Lviv (2015). · Zbl 1338.15034 [11] Chen, S.; Tian, Y., On solutions of generalized Sylvester equation in polynomial matrices, J. Franklin Inst., 351, 12, 5376-5385 (2014) · Zbl 1398.65078 [12] Martins, F.; Pereira, E., Block matrices and stability theory, Tatra Mt. Math. Publ., 38, 147-162 (2007) · Zbl 1199.34264 [13] Newman, M., The Smith normal form of a partitioned matrices, J. Res. Natl. Bur. Stand., 78B, 1, 3-6 (1974) · Zbl 0281.15009 [14] Petrychkovych, V.; Dzhaliuk, N., Factorizations in the rings of the block matrices, Bul. Acad. Ştiinţe Repub. Mold. Mat., 85, 3, 23-33 (2017) · Zbl 1396.15012 [15] V. Shchedryk, Arithmetic of Matrices over Rings, Akademperiodyka, Kyiv (2021). This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.