Equivalence of matrices in the ring $M(n, R)$ and its subrings
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 $M_{BT} (n_1, . . . , n_k, R)$.
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