Equivalence of matrices in the ring M(n,R) and its subrings
DOI:
https://doi.org/10.37863/umzh.v73i12.6858Keywords:
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 MBT(n1,...,nk,R) and block diagonal matrices MBD(n1,...,nk,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 MBT(n1,...,nk,R) of block triangular matrices if and only if their main diagonals are equivalent in the subring MBD(n1,...,nk,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 MBT(n1,...,nk,R), then these matrices are equivalent in the subring MBT(n1,...,nk,R).
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