An algorithmic version of the theorem by Latimer and MacDuffee for 2×2 integral matrices Original Research Article

Given two n×n integral matrices A and B, they are said to be equivalent if B=S−1AS, where S is an n×n integral matrix with determinant ±1. If we consider n×n integral matrices with a fixed characteristic polynomial that is irreducible over image , it is well known from a result by Latimer and MacDuffee that the number of matrix classes (equivalence classes of matrices) is equal to the number of ideal classes (I congruent with J if I=qJ for some q in the quotient field) of the ring obtained by adjoining a root of the characteristic polynomial to image . In this paper, we develop an effective version of this result for 2×2 matrices. We present an algorithm which given a 2×2 matrix finds a canonical representative in its class. In particular this allows us to determine whether two matrices are equivalent.