An algorithmic version of the theorem by Latimer and MacDuffee for 2\({\times}\)2 integral matrices
The matrix solutions of an algebraic equation \(f(x) \equiv x^n + k_1x^{n-1}+ \cdots +k_n=0\), with integral coefficients, that is irreducible over \(\mathbb{Q}\), were studied earlier by \textit{C. G. Latimer} and \textit{C. C. MacDuffee} [Ann. Math., II. Ser. 34, 313-316 (1933; Zbl 0006.29002)]. If a matrix \(A\) is the solution then all matrices of the class \(S^{-1}AS\) will again be solution if \(S\) is an \(n \times n\) integral matrix with determinant \(\pm 1\). In general, all solutions cannot be derived in this way from one solution only. The known result by Latimer and MacDuffee is that the number of classes of matrix solutions coincides with the number of different classes of ideals in the ring obtained by adjoining a root of the algebraic equation to \(\mathbb{Z}\). This result is here modified into an algorithmic version for \(2 \times 2\) matrices. The algorithm enables to determine if two matrices are equivalent and also allows to find a canonical representative for a \(2 \times 2\) matrix.
scientific article; zbMATH DE number 1758605