Equivalence of Matrices in the Ring M(n, R) and its Subrings

摘要

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), . . . , n(k), R) and block diagonal matrices M-BD (n(1), . . . , 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), . . . , 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 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).