# Equivalence of matrices in the ring $M(n, R)$ and its subrings

• N. S. Dzhaliuk Pidstryhach Institute for Applied Problems of Mechanics and Mathematics National Academy of Sciences of Ukraine
• V. M. Petrychkovych Pidstryhach Institute for Applied Problems of Mechanics and Mathematics National Academy of Sciences of Ukraine
Keywords: ring of matrices, subring of block triangular matrices, subring of block diagonal matrices, еquivalence, the Smith normal form.

### Abstract

UDC 512.64+512.55

In this article, we consider the equivalence of matrices in the ring $M(n, R)$ and in its subrings of block triangular matrices $M_{BT} (n_1, . . . , n_k, R)$ and block diagonal matrices $M_{BD} (n_1, . . . , n_k, R)$ where $R$ is a commutative principal ideal domain, and investigate the connections between these equivalences. Under the conditions that the block triangular matrices are block diagonalizable, i.e., equivalent to their main block diagonals, we establish that these matrices are equivalent in the ring $M_{BT} (n_1, . . . , n_k, R)$ of block triangular matrices if and only if their main diagonals are equivalent in the subring $M_{BD} (n_1, . . . , n_k, R)$ 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 the Smith normal forms $S(A) = S(B)$ are equivalent to the Smith normal forms in the subring $M_{BT} (n_1, . . . , n_k, R)$, then these matrices are equivalent in the subring $M_{BT} (n_1, . . . , n_k, R)$.

### References

A. Dmytryshyn, B. K˚agstr¨om, Coupled Sylvester-type matrix equations and block diagonalization, SIAM J. Matrix Anal. and Appl., 36, № 2, 580 – 593 (2015); https://doi.org/10.1137/151005907 DOI: https://doi.org/10.1137/151005907

W. E. Roth, The equations $AX-YB=C$ and $AX-XB=C$ in matrices, Proc. Amer. Math. Soc., 3, 392 – 396 (1952); https://doi.org/10.2307/2031890 DOI: https://doi.org/10.1090/S0002-9939-1952-0047598-3

R. B. Feinberg, Equivalence of partioned matrices, J. Res. Natl. Bur. Stand., B, 80B, № 1, 89 – 97 (1976). DOI: https://doi.org/10.6028/jres.080B.015

W. H. Gustafson, Roth’s theorem over commutative rings, Linear Algebra and Appl., 23, 245 – 251 (1979); https://doi.org/10.1016/0024-3795(79)90106-X DOI: https://doi.org/10.1016/0024-3795(79)90106-X

N. S. Dzhaliuk, V. M. Petrychkovych, Solutions of the matrix linear bilateral polynomial equation and their structure, Algebra Discrete Math., 27, № 2, 243 – 251 (2019).

V. M. Bondarenko, Zobrazhennia helfandovykh hrafiv, Pratsi In-t matematyky NAN Ukrainy, Kyiv (2005).

V. V. Sergeichuk, Canonical matrices and related questions, Pratsi In-t matematyky NAN Ukrainy, Kyiv , 57 (2006).

I. Gohberg, P. Lancaster, L. Rodman, Matrix polynomials, Academic Press, New York (1982).

V. M. Petrychkovich, Cell-triangular and cell-diagonal factorizations of cell-triangular and cell-diagonal polynomial matrices, Math. Notes., 37, № 6, 431 – 435 (1985). DOI: https://doi.org/10.1007/BF01157677

V. M. Petrychkovich, Uzahalnena ekvivalentnist matryts i yikh naboriv ta faktoryzatsiia matryts nad kiltsiamy, In-t prykl. probl. mekhaniky i matematyky NAN Ukrainy, Lviv (2015).

S. Chen, Y. Tian, On solutions of generalized Sylvester equation in polynomial matrices, J. Franklin Inst., 351, № 12, 5376 – 5385 (2014); https://doi.org/10.1016/j.jfranklin.2014.09.024 DOI: https://doi.org/10.1016/j.jfranklin.2014.09.024

F. Martins, E. Pereira, Block matrices and stability theory, Tatra Mt. Math. Publ., 38, 147 – 162 (2007).

M. Newman, The smith normal form of a partitioned matrices, J. Res. Natl. Bur. Stand., B. Math. Sci., 78B, № 1, 3 – 6 (1974). DOI: https://doi.org/10.6028/jres.078B.002

V. Petrychkovych, N. Dzhaliuk, Factorizations in the rings of the block matrices, Bul. Acad. ¸ Stiin¸te Repub. Mold. Mat., 85, № 3. 23 – 33 (2017).

V. Shchedryk, Arithmetic of matrices over rings, Pidstryhach Inst. Appl. Probl. Mech. Math. of NAS of Ukraine, Akademperiodyka, Kyiv (2021). DOI: https://doi.org/10.15407/akademperiodika.430.278

Published
17.12.2021
How to Cite
Dzhaliuk , N. S., and V. M. Petrychkovych. “Equivalence of Matrices in the Ring $M(n, R)$ and Its Subrings”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 73, no. 12, Dec. 2021, pp. 1612 -18, doi:10.37863/umzh.v73i12.6858.
Issue
Section
Research articles