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Greatest common divisors of Euclidean domain matrices. (English) Zbl 1375.15029

We recall that a Euclidean domain benefits from having no zero divisors, leading to an analogous Euclidean algorithm, with quotient and remainder and greatest common divisor property. In this paper, the authors consider the problem of determining a greatest common divisor of two Euclidean domain matrices, those whose entries entries lie in a Euclidean domain (also known as a Euclidean ring). The main focus of this paper is to extend the greatest common divisor property to Euclidean domain matrices.
The usual definitions for three matrices \(A\), \(B\) ,\(C\) satisfying \(A=BC\) are employed, so that the matrices \(B\) and \(C\) are respectively referred to as left and right divisors of the matrix \(A\). Additional definitions include denoting by \(E^{m\times n}\), the set of all \(m\times n\) matrices with entries in the Euclidean domain \(E\); by deg \(A=r\), with \(A\in E^{m\times n}\), and \(r\leq\min(m,n)\), the maximum degree \(r\), of all of the maximum order minors of \(A\), and that \(A\in E^{n\times n}\) is unimodular if there exists a matrix \(\hat{A}\in E^{n\times n}\) such that \(A\hat{A}=\hat{A}A=I_n\), with \(I_n\) the \(n\times n\) identity matrix.
Using the Smith form \(S_A=(s_{i j})_{m\times n}\) of a matrix \(A\in E^{m\times n}\), so that for \(f_i=s_{i i}\in S_A\), we then have that (Theorem 7) \(f_i\) divides \(f_{i+1}\), \(1\leq i\leq r-1\). The Smith form enables the definition of a unimodular Euclidean domain matrix to be extended from square matrices to left and right unimodular \(m\times n\) matrices. From this, it is shown (Theorem 10) that every Euclidean domain matrix \(A\in E^{m\times n}\) of rank \(A=r\), can be factorised (non-uniquely) as \[ A=A_L' A_1,\quad \text{or}\quad A=\hat{A_1}A'_R, \] where \(A_1\in E_{r\times n}\) is right unimodular and \(\hat{A}_1\in E_{m\times r}\) is left unimodular.
A greatest common left divisor \(B\), of two Euclidean domain matrices \(A_1\) and \(A_2\) is then established (Theorem 14), where if \(C\) is any other greatest common divisor matrix of \(A_1\) and \(A_2\), then there exists a unimodular matrix \(U\), with \(C=BU\). This result facilities the definition of two matrices \(A_1\) and \(A_2\) being left coprime if their greatest common left divisor is unimodular, and similarly for the right coprime definition.
Five equivalent statements are then deduced (Theorem 16), which say that for \(A_1\in E^{m\times n_1}\), \(A_2\in E^{m\times n_2}\), with \(n_1+n_2=n\geq m=\text{rank}(A_1\,\,A_2)\), i.e., the rank of the \(m\times n\) matrix \((A_1\,\,A_2)\), then
\(A_1\) and \(A_2\) are co-prime, so that their greatest common divisor matrix \(B\) is unimodular,
the \(m\times n\) Euclidean domain matrix \(B=(A_1\,\, A_2)\) has no zeros in \(\mathbb{C}\) (see the paper for further details),
there exists a unimodular matrix \(\overline{A}_R\in E^{n\times n}\), such that \[ (A_1\,\, A_2)\overline{A}_R=(I_m\,\,0_{m,n-m})=S_A, \]
there exists matrices \(Y\in E^{n_1\times m}\) and \(Z\in E^{n_2\times m}\), such that \(A_1 Y+A_2 Z=I_m\), and
there exists matrices \(A_3\in E^{(n-m)\times n_1}\) and \(A_4\in E^{(n-m)\times n_2}\), such that the square matrix \[ \begin{pmatrix} A_1 & A_2\\ A_3 & A_4 \end{pmatrix}\in E^{n\times n}, \] is unimodular.

MSC:

15A30 Algebraic systems of matrices
13F07 Euclidean rings and generalizations
15A23 Factorization of matrices
15A21 Canonical forms, reductions, classification
11A05 Multiplicative structure; Euclidean algorithm; greatest common divisors
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