TY - JOUR
AU - N. S. Dzhaliuk
AU - V. M. Petrychkovych
PY - 2021/12/17
Y2 - 2023/09/25
TI - Equivalence of matrices in the ring $M(n, R)$ and its subrings
JF - Ukrainsâ€™kyi Matematychnyi Zhurnal
JA - Ukr. Mat. Zhurn.
VL - 73
IS - 12
SE - Research articles
DO - 10.37863/umzh.v73i12.6858
UR - https://umj.imath.kiev.ua/index.php/umj/article/view/6858
AB - UDC 512.64+512.55In 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 $M_{BT} (n_1, . . . , n_k, R)$.
ER -