Equivalence of pairs of matrices with relatively prime determinants over quadratic rings of principal ideals (Q2831422)

Let \(\mathbb{Z}\) be the ring of integers, and \(\mathbb{K}=\mathbb{Z} [\sqrt{k} ]\) a quadratic ring, where \(k\neq 0,1\) is a square-free element of \(\mathbb{Z}\).NEWLINENEWLINEIn this paper, the authors investigate the equivalence of matrices and their pairs, defined such that NEWLINE\[NEWLINE A\rightarrow UAV^A,\qquad (A,B)\rightarrow (UAV^A,UBV^B), NEWLINE\]NEWLINE where \(U\in \text{GL}(n,\mathbb{Z})\), and \(V^A,\,V^B\in \text{GL}(n,\mathbb{Z} [\sqrt{k} ])\).NEWLINENEWLINEMore specifically, they establish that if a pair of matrices \(A,\,B\) have relatively prime determinants over the quadratic principal ideal ring, then they can be reduced by means of such equivalent transformations to the pairs \(T^A,\,T^B\) of triangular forms with invariant factors on the main diagonal.NEWLINENEWLINEThis result implies that given such a pair of matrices \(A,\,B\), with \((\text{det}\,A,\,\text{det}\,B)=1\), then there exists the invertible matrices \(U\in \text{GL}(n,\mathbb{Z})\), and \(V^A,\,V^B\in \text{GL}(n,\mathbb{Z} [\sqrt{k} ])\), with NEWLINE\[NEWLINE UAV^A\rightarrow T^A,\qquad UBV^B\rightarrow T^B, NEWLINE\]NEWLINE where \(T^A\) and \(T^B\) are lower triangular matrices whose principal diagonal entries comprise the invariant factors.